QUALITY INDEX AND KINEMATIC ANALYSIS OF SPATL\L REDUNDANT IN-PARALLEL MANIPULATORS i - ' - :■' *: '' if:' ^K By YU ZHANG A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2000 Copyright 2000 by Yu Zhang To my wife, Ying, and our parents. ACKNOWLEDGMENTS I want to express my deep and sincere gratitude to Dr. Joseph Duffy, my supervisor during my Ph.D. study, for providing me with the opportunity to complete my study under his exceptional guidance. Without his untiring patience, constant encouragement, guidance and knowledge this work would not have been possible. I would also like to thank my supervisory committee members. Dr. Carl D. Crane, Dr. Gloria J. Wiens, Dr. Ali A. Seirig, and Dr. Ralph Selfridge. I am grateful for their willingness to serve on my committee, providing me help whenever needed and for reviewing this dissertation. I especially thank Professor Chonggao Liang of Beijing University of Posts and Telecommunications for educating me on the various aspects of mechanism analysis and design. Also, I would like to thank all my colleagues in the Center for Intelligent Machines and Robotics for their help and support. Finally, I would like to thank my lovely wife, Ying Zhu. Her love, support and encouragement has had made my life rich and complete. I am grateful to my parents and parents-in-law for their constant support and encouragement throughout my educational endeavors. IV TABLE OF CONTENTS ACKNOWLEDGMENTS iv LIST OF TABLES vii LIST OF FIGURES viii ABSTRACT xi L INTRODUCTION 1 1.1 Redundant Parallel Manipulators 1 1.2 Quality Index 6 1.3 Outline of Dissertation 9 2. SPATL\L GEOMETRY AND STATICS 12 2.1 Pliicker Line Coordinates 12 2.2 Statics of a Rigid Body 15 2.3 The Statics of a Parallel Manipulator 20 3. THE OPTIMUM QUALITY INDEX FOR A SPATIAL REDUNDANT 4-4 IN-PARALLEL MANIPULATOR 24 3.1 Determination of -ydet J^J^ 25 3.2 Implementation 30 4. THE KINEMATIC ANALYSIS OF THE SPATIAL REDUNDANT 4-4 IN-PARALLEL MANIPULATOR 43 4.1 Inverse Kinematic Analysis 44 4.2 Forward Kinematic Analysis 47 4.2.1 Introduction 47 4.2.2 Coordinate Transformations 48 4.2.3 Constraint Equations 51 4.2.4 The Solution 52 4.2.5 Numerical Verification 55 5. THE OPTIMUM QUALITY INDEX FOR A SPATIAL REDUNDANT 4-8 IN-PARALLEL MANIPULATOR 58 5.1 Determination of ydet J^J^ 59 5.2 Implementation 64 6. THE FORWARD KINEMATIC ANALYSIS OF THE SPATIAL REDUNDANT 4-8 IN-PARALLEL MANIPULATOR 79 6.1 Forward Kinematic Analysis 79 6.2 Numerical Verification 82 7. THE OPTIMUM QUALITY INDEX FOR A SPATIAL REDUNDANT 8-8 IN-PARALLEL MANIPULATOR 85 7.1 Determination of ^det J^J^ 85 7.2 Implementation 94 8. THE FORWARD KINEMATIC ANALYSIS OF THE SPATL\L REDUNDANT 8-8 IN-PARALLEL MANIPULATOR 1 10 8.1 Coordinate Systems 110 8.2 Constraint Equations 1 13 8.3 Equation Solution 1 14 8.4 Numerical Verification 118 9. CONCLUSIONS 121 APPENDIX A: CONSTANTS FOR THE FORWARD KINEMATIC ANALYSIS OF THE REDUNDANT 4-4 IN-PARALLEL MANIPULATOR 123 APPENDIX B: CONSTANTS FOR THE FORWARD KINEMATIC ANALYSIS OF THE REDUNDANT 8-8 IN-PARALLEL MANIPULATOR 130 LIST OF REFERENCES 148 BIOGRAPHICAL SKETCH 152 VI LIST OF TABLES Table Page Table 4.1: Numerical results of the redundant 4-4 in-parallel manipulator 56 Table 4.2: A numerical example for the special case of the redundant 4-4 in-parallel manipulator 57 Table 6. 1 : Numerical results of the redundant 4-8 in-parallel manipulator 83 Table 6.2: A numerical example for the special case of the redundant 4-8 in-parallel manipulator 84 Table 8.1: Numerical results of the redundant 8-8 in-parallel manipulator 119 Table 8.2: A numerical example for the special case of the redundant 8-8 in-parallel manipulator 119 vu LIST OF FIGURES Figure Page Figure 1.1: A planar parallel x-y manipulator with one redundant actuator 2 Figure 1.2: A 2-DoF planar parallel manipulator 3 Figure 1.3: A redundant 2-DoF planar parallel manipulator 3 Figure 1.4: Planar view of spatial nonredundant 4-4 in-parallel manipulators 4 Figure 1 .5: Self-deployable space structure 1 1 Figure 2.1: Determination of a line 13 Figure 2.2: Pliicker line coordinates 15 Figure 2.3: Representation of a force on a rigid body 16 Figure 2.4: Dyname and wrench 19 Figure 2.5: A 6-6 in-parallel manipulator 21 Figure 3.1: A redundant 4-4 in-parallel manipulator 24 Figure 3.2: Plan view of the redundant 4-4 in-parallel manipulator 24 Figure 3.3: Plan view of the optimal configuration of the redundant 4-4 in-parallel manipulator with the maximum quality index 30 Figure 3.4: Quality index for platform vertical movement 31 Figure 3.5: Quality index for platform horizontal translation 34 Figure 3.6: Platform rotations about the ^''-axis 35 Figure 3.7: Quality index for platform rotations about the a:'- and >''-axes 37 Figure 3.8: Platform rotations about the z-axis 38 Vlll Figure 3.9: Quality index for platform rotation about the z-axis 40 Figure 3.10: Plan view of the singularity position of the redundant 4-4 in-parallel manipulator when 6z = 90° 41 Figure 4.1: Coordinate systems of a redundant 4-4 in-parallel manipulator 45 Figure 4.2: Coordinate transformations 49 Figure 5. 1 : A redundant 4-8 in-parallel manipulator 58 Figure 5.2: Plan view of the redundant 4-8 in-parallel manipulator 58 Figure 5.3: Plan view of the optimal configuration of the redundant 4-8 in-parallel manipulator with the maximum quality index 63 Figure 5.4: Compatibility between the redundant 4-4 and the 4-8 parallel manipulators. 64 Figure 5.5: Quality index for platform vertical movement 65 Figure 5.6: Quality index for platform horizontal translation with different values of j5. 70 Figure 5.7: Platform rotation about they '-axis 71 Figure 5.8: Quahty index for platform rotations about the jc'- and j'-axes 73 Figure 5.9: Platform rotation about the z-axis 74 Figure 5.10: Quality index for platform rotation about the z-axis 76 Figure 5.1 1: Plan view of the singularity position of the redundant 4-8 in-parallel manipulator when ^^ = 90° 77 Figure 6.1: Coordinate systems of a redundant 4-8 in-parallel manipulator 80 Figure 6.2: Leg relations 81 Figure 7.1: A redundant 8-8 in-parallel manipulator 86 Figure 7.2: Plan view of the redundant 8-8 in-parallel manipulator 86 Figure 7.3: Plot of /(a, /3) = 2aj3-2a-2/3 + \ = 90 Figure 7.4: Plot of /i vs. aand yff with a = 1 92 Figure 7.5: Plot of V^et J^ J^ vs. av/kha= 1 93 Figure 7.6: An example of redundant 8-8 manipulator in optimal configuration... 93 IX Figure 7.7: Quality index for platform vertical movement 96 Figure 7.8: Reduction of the size of the redundant 8-8 in-parallel manipulator 96 Figure 7.9: Quality index for platform horizontal translation with different values of alOO Figure 7.10: Platform rotations about the >' '-axis 102 Figure 7.1 1: Quality index for platform rotations about the x'- and >' '-axes 105 Figure 7.12: Platform rotations about the z-axis 106 Figure 7.13: Quality index for platform rotation about the z-axis 108 Figure 7.14: Plan view of the singularity position of redundant 8-8 in-parallel manipulator when 6^ = 90° 109 Figure 8.1: Coordinate systems of a redundant 8-8 in-parallel manipulator Ill Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy QUALITY INDEX AND KINEMATIC ANALYSIS OF SPATIAL REDUNDANT IN-PARALLEL MANIPULATORS By Yu Zhang , ,1 December 2000 ^ ^■>. t Chairman: Dr. Joseph Duffy • ' - Major Department: Mechanical Engineering Parallel manipulators have been the subject of much investigation over the last decade because of their inherent advantages of load carrying capacity and spatial rigidity compared to serial manipulators. Usually they have the same number of actuators as their degree of freedom, but in some cases, it may be interesting to have more actuators than needed and to consider redundant parallel manipulators. Redundancy in actuation can be used to increase dexterity, to reduce or even eliminate singularities, to increase reliability, to simplify the forward kinematics, and to improve load distribution in actuators. The purpose of this work is to design and analyze several spatial redundant parallel manipulators. The proposed quality index will assist a designer to choose the relative dimensions of the fixed and moving platforms, locate joint centers in the fixed and moving platforms, determine an optimum position which would be an 'ideal' location of XI the workspace center, and determine acceptable ranges of pure translations and pure rotations for which the platform is stable. The quality index for redundant parallel manipulators is defined as a dimensionless ratio that takes a maximum value of 1 at a central syimnetrical configuration that is shown to correspond to the maximum value of the square root of the determinant of the product of the manipulator Jacobian by its transpose. The Jacobian matrix is none other than the normalized coordinates of the leg lines. When the manipulator is actuated so that the moving platform departs from its central configuration, the determinant always diminishes, and, as is well known, it becomes zero when a special configuration is reached (the platform then gains one or more uncontrollable freedoms). It is shown that the quality index A, for which 0<A< 1, can be used as a constructive measure of not only acceptable and optimum design proportions but also an acceptable operating workspace (in the static stability sense). We also studied the forward kinematic analysis of the redundant in-parallel manipulators to determine the position and orientation of the platform, given the leg lengths. xu CHAPTER 1 INTRODUCTION Parallel manipulators have been studied extensively over the last decade with their high structural stiffness, position accuracy and good dynamic performance. Usually they have the same number of actuators as their degree of freedom, but in some cases, it may be interesting to have more actuators than needed to overcome disadvantages of the nonredundant parallel manipulators shown by Merlet [25]. 1 . 1 Redundant Parallel Manipulators A number of redundant parallel manipulators have been studied in literature, for example, the development of a direct-drive redundant parallel manipulator for haptic displays by Buttolo and Hannaford [2, 3], the design of a 2-DoF parallel manipulator (Figure 1.1) with actuation redundancy for high speed and stiffness-controlled operation by Kock and Schumacher [17], and the addition of a redundant (fourth) branch to three- branch manipulators for the purpose of uncertainty elimination and assembly mode reduction by Notash and Podhorodeski [28]. Maeda et al. [24] also designed a redundant wire-driven parallel manipulator that is suitable especially for high speed assembling of lightweight objects such as semiconductors. By studying a parallel machining center, O'Brien and Wen [29] examined the effectiveness of singularity modification through redundant actuation and suggested that augmenting the actuation of a mechanism provides a mechanically feasible means of increasing kinematic manipulability. Figure 1.1: A planar parallel x-y manipulator with one redundant actuator (Adapted from: Kock and Schumacher, A Parallel X-y Manipulator with Actuation Redundancy for High Speed and Active Stiffness Applications (1998) [17]) Leguay-Durand and Reboulet [23] studied a redundant spherical parallel manipulator and showed that actuator redundancy removes singularities and improves dexterity in an enlarged workspace. Using a conditioning measure, they compared the redundant spherical parallel manipulator with an equivalent nonredundant structure and found notably improved uniformity of dexterity for the redundant structure. Similar results were also found by Kurtz and Hay ward [19]. Kokkinis and Millies [18] found that actuation redundancy allows the selection of optimal joint torque for a given load. Nakamura and Ghodoussi [27] also showed that the redundant actuation could increase the payload and improve the dynamic response of manipulators. Dasgupta and Mruthyunjaya [6, 7] saw the redundancy of parallel manipulators as the series-parallel dual part of redundancy in serial manipulators. They proposed the concept of force (static) redundancy for redundancy in parallel manipulators in contrast to kinematic redundancy (widely studied in literature) in serial manipulators. In summary, redundant parallel manipulators have the following advantages: 1. Redundancy in actuation can be used to increase dexterity and reduce or even eliminate singularities of parallel manipulators (Pemg and Hsiao [30]). Usually, parallel manipulators have a high stiffness, except in some special positions or postures where the platform has self-motion and may even collapse. These singular configurations may cause serious damage to the manipulator and/or objects in its environment. Redundant legs can be used to pull out the platform from singularity positions. For example, for the 2-DoF planar parallel manipulator shown in Figure 1.2, its singularity positions can be found on the line joining the two fixed pivot positions, and this singularity can be eliminated by adding another leg as shown in Figure 1.3. The new redundant parallel manipulator is entirely free from singularity as long as its three fixed pivots are noncollinear. |6C=^ Figure 1 .2: A 2-DoF planar parallel Figure 1 .3: A redundant 2-DoF planar manipulator parallel manipulator Here is another example, considering the two cases of spatial nonredundant 4-4 in-parallel manipulators shown in Figure 1 .4. They are in singularity posifions when their platforms are parallel to the base. Such singularity is dangerous because it is not immediately obvious from its configuration and if we build such manipulators, they may collapse immediately when their platforms are parallel to the base. One possible solution is to add another two legs to form a redundant 4-4 in-parallel manipulator (Chapters) (Figure 3.1). (a) (b) Figure 1.4: Planar view of spatial nonredundant 4-4 in-parallel manipulators 2. Redundancy in actuation can be used to increase the reliability of in-parallel manipulators (Shin and Lee [33]). That is, even if some of the actuators fail, a manipulator can still operate normally as long as the number of operating actuators is not less than the mobility of the manipulator. Thus such a redundant system has a failure tolerance, which is increasingly important in robotics, especially when robots and manipulators are used in remote or harsh environments such as space, deep sea, nuclear plants and for bomb disposal. Because these environments do not allow ~H~5:/"''<'^ immediate human intervention for repair or recovery, the ability of a robot or a manipulator to cope with the failures becomes desirable. 3. The information from the length of the redundant legs can be used to simplify the forward kinematics. When controlling a parallel manipulator, we need to do the forward kinematic analysis, i.e., to determine the configuration of the moving platform given all the leg lengths. This analysis is usually difficult as it involves a set of nonlinear equations and, generally, there is more than one solution. For example, the forward analysis for the general 6-6 parallel manipulator requires the solution of a 40"^ degree polynomial (Raghavan [31]) the solution of which is clearly impractical for real-time implementation. The additional information from the redundant legs reduces many uncertainty positions and even can obtain a unique solution to the forward analysis. 4. The actuator forces and joint torques in the redundant parallel manipulators are not uniquely determined. This characteristic can be used to optimize some criteria. For example, the joint torque required for a given motion can be minimized. Accordingly, it is possible to increase the payload of a closed-link mechanism by adding redundant actuators. Some other advantages of using redundant actuators are increasing workspace while improving dexterity, having autonomous calibration, and building variable geometry trusses. Possibilities of redundancy in parallel manipulators and their effective use have not been studied extensively until now. The purpose of this work is to design and analyze several spatial redundant parallel manipulators. 1.2 Quality Index Parallel manipulators have better load carrying capacity and spatial rigidity than serial manipulators. However, the complexity of the kinematics of parallel manipulators makes it more difficult for a designer to determine a set of kinematic and geometry parameters that will efficiently produce prescribed performances. Indeed, the behavior of parallel manipulators is far less intuitive than that of serial manipulators. The geometric properties associated with singularities, for example, may be much more difficult to identify directly (Fichter [10] and Merlet [26]). Therefore, more systematic analysis and optimization tools are needed to make parallel manipulators more accessible to designers. At this time little information is available to assist designers in the following task: (a) Choose the relative sizes of the fixed and moving platforms. (b) Locate the positions of the centers of the spherical joints in the base and the centers in the moving platform. (c) Determine an optimum position that would be an ideal 'center' location of the workspace. (d) Determine acceptable ranges of pure translations of the platform for which the platform is stable (i.e., not too close to a singularity). However, the question "How close is too close?" is often hard to answer. (e) Determine acceptable ranges of pure rotations of the platform for which the platform is stable. (f) Determine the ranges of leg displacements. These considerations are the reasons that the quality index was proposed. The quality index was defined initially for a planar 3-3 in-parallel device by the dimensionless ratio (Lee, Duffy, and Keler [22]) > }'.- ; ■.■ , detj >^ = i' 'i (1.1) where J is the three-by-three Jacobian matrix of the normalized coordinates of three leg lines. Then it was defined for an octahedral in-parallel manipulator by Lee, Duffy, and Hunt [21] and 3-6, 6-6 in-parallel devices by Lee and Duffy [20]. For these cases J is the six-by-six matrix of the normalized coordinates of the six leg lines. For these fully synmietrical nonredundant parallel manipulators the quality index takes a maximum value of /I = 1 at a central symmetrical configuration that corresponds to the maximum value of the determinant of the six-by-six Jacobian matrix (i.e., det J = det 7m) of the manipulator. When the manipulator is actuated so that the moving platform departs from its central configuration, the determinant always diminishes, and, as is well known, it becomes zero when a special configuration is reached (the platform then gains one or more uncontrollable freedoms). In this dissertation, the quality index is extended for redundant manipulators by the dimensionless ratio detJJ^ VdetJ^J, ' m" m This makes complete sense because the Cauchy-Binet theorem det JJ^ =A^ +A2 -I---+A^ , has geometrical meaning. Here, each A, ( 1 < / < m = ) is simply the determinant of the 6x6 submatrices of J which is a 6xn matrix. This is clear when w = 6, (1.2) reduces to (1.1). It has been shown by Lee et al. [21] that by using the Grassmann-Cayley algebra (White and Whiteley [35]), for a general octahedron, when the leg lengths are not normalized, det J has dimension of (volume)^ and it is directly related to the products of 8 volumes of tetrahedra that form the octahedron. In this way detj and vdet JJ^ have geometrical meaning. We mention in passing the work of Cox [4] and Duffy [8], both of which cover special configurations of planar motion platforms. Hunt and McAree [14] go into considerable detail regarding the general octahedral manipulator. Its special configurations are described in the context of other geometrical properties. A few papers were published on the optimal design of nonredundant parallel manipulations (see for example Gosselin and Angeles [11, 12], Zanganeh and Angeles [36]). Zanganeh and Angeles [36] point out problems with quantities such as condition number due to the inherent inhomogeneity of the columns of the Jacobian, /. This is precisely why equations (1.1) and (1.2) are adopted as an index of quality rather than other well-established methods (found in books on theory of matrices and linear algebra) that lead (via norms, diagonalization and singular values, etc.) to properties that relate to 'conditioning'. All such methods are based implicitly on the presumption that a column- vector (say, of a six-by-six matrix) can be treated as a vector in 9?^. However, the six elements in the column of a typical robot Jacobian are the normalized coordinates of a screw (almost always of zero pitch; i.e., a line); in a metrical coordinate frame three of them are dimensionless and three have dimension [length], such a length being the measure of the moment about a reference point of a unit force. The column generally comprises two distinct vectors (each of them in 91^). For the legs of the nonredundant and redundant manipulators it is not possible to remove all the length dimensions from their coordinates. Even the adoption of some artificial length unit fails, simply because a moment can never be converted to a pure force. Moreover, any index of quality derived from such textbook techniques is likely to vary according to the coordinate frame in which the Jacobian is formulated. Our method works for two reasons: first, the determinant of a (square) Jacobian of line coordinates depends solely on the configuration in 9t^ of the actuated axes and not on the coordinate frame in which the line coordinates are determined. The second reason is that equations (1.1) and (1.2) are dimensionless ratios, and our quality indices are always independent of the choice of units of length measurement. Unlike the case of a mechanism designed for a specific task, the tasks to be performed by a manipulator are varied. Hence, there should not be any preferred general orientation for which the manipulator would have better properties. It suggests that the manipulator should be symmetrical. Such symmetrical configurations may not always exist, of course. However, except for unusual applications (and there will undoubtedly be some where for example unusual loads must be sustained) we are safe in seeking centrally symmetrical designs to which we can assign the highest quality index /l=l, or close to it. For these cases, contours of quality index help to determine a realistic workspace volume that is free from singularities. Therefore, we are concerned primarily with symmetrical redundant parallel manipulators in this dissertation. 1 .3 Outline of Dissertation A simple introduction to the screw theory is presented in Chapter 2 to provide insight into how a screw-based Jacobian matrix of a parallel manipulator is determined. In Chapters 3 and 4 a spatial redundant 4-4 in-parallel manipulator is studied first. The device consists of a square platform and a square base connected by eight actuated legs. As in Chapter 3, the quality index for the redundant 4-4 parallel manipulator is 10 determined. To achieve the maximum quality index for a redundant 4-4 in-parallel manipulator with platform side a, the base has side 42a and the perpendicular distance between the platform and the base is —j= . The kinematic analysis of the redundant 4-4 V* in-parallel manipulator is studied in Chapter 4. The derivation of forward kinematic equations for position and orientation of the platform is described. Chapters 5 and 6 extend the study to a redundant 4-8 in-parallel manipulator with a square platform and an octagonal base. The octagonal base is formed by separating from each vertex of a square by a small distance. The quality index for this manipulator is determined in Chapter 5. The compatibility between the redundant 4-4 and the 4-8 parallel manipulators also is discussed in this Chapter. Chapter 6 solves the forward kinematics of the redundant 4-8 parallel manipulator by transferring the problem to the corresponding redundant 4-4 case. . Finally, in Chapters 7 and 8, a redundant 8-8 in-parallel manipulator is studied. The device has an octagonal platform and a similar octagonal base connected by eight legs. Such arrangement avoids using double-spherical joints because they can produce serious mechanical interference. However, by using the quality index determined in Chapter 7, the best design can be obtained when the pair of separated joints in the base and top platform are as close as possible. In Chapter 8, the kinematic analysis of the redundant 8-8 parallel manipulator is performed. The forward analysis gives a much simpler solution than that of the nonredundant case. Using quality index, variable motions are investigated for which a moving platform rotates about a central axis or moves parallel to the base. The quality index can be used as a constructive measure not only of acceptable and optimum design proportions 11 but also of an acceptable operating workspace (in the static stability sense). Moreover, analysis of these redundant in-parallel manipulators can be used to model and design a self-deployable space structure that has a pair of flexible antenna platforms in the base and top platform as shown in Figure 1.5 (Duffy et al. [9] and Knight et al. [16]). Figure 1.5: Self-deployable space structure CHAPTER 2 SPATIAL GEOMETRY AND STATICS Chapter 1 showed that the quaUty index of parallel manipulators is based on the Jacobian matrix. This chapter, which is mostly a general background in screw theory (Ball [1]), provides insight into how the Jacobian matrix of parallel manipulators is determined. Firstly, we review some basic concepts of spatial geometry and screw theory. 2. 1 Pliicker Line Coordinates Two distinct points ri(A:i, y\, z\) and r2(x2, yi, zi) can be connected by a line in space. The vector S whose direction is along the line can be written in the form S = r2-ri. (2.1) Alternatively this may be expressed as S = Li + Mj+Nk (2.2) where L = X2-xi, M = y2-y\, N = Z2-Z\ (2.3) are defined as the direction ratios of the line and they are related to the distance ISI between the two points by L^ + M^ + N^ = \S\^ (2.4) where the notation 1 1 denotes absolute magnitude. Often L, M, and A^ are expressed in the form L= —^ -, M = ^^ — ^, N=— -, (2')) ISI ISI ISI ^"^-^^ which consists of unit direction ratios of the line, and (2.4) reduces to 12 13 l^ + m'^ + n^=\. (2.6) If r represents a vector from the origin to any general point on the Une (Figure 2.1), then the vector r-ri is parallel to S and therefore the equation of the line can be written as (r-ri)xS = (2.7) and in the form where r X S = So So = ri X S (2.8) (2.9) is the moment of the line about origin O and is clearly origin dependent. Further, because So=rixS, the vectors S and S© are perpendicular and as such satisfy the orthogonality condition S • So = 0. (2.10) Figure 2. 1 : Determination of a line 14 The coordinates of a line are written as [S; So]' and are referred to as the Plucker coordinates of the Hne [13]. The coordinates [S; So] are homogeneous since from (2.8) the coordinates [kS; kSo] (k is a non-zero scalar) determine the same line. Expanding (2.9) yields • I • J k X\ y^ Z\ L M N So = which can be expressed in the form So = Pi + OJ + ^k where (2.11) (2.12) P= y,N - z,M, Q= z^L - x^N, R= XiM - y,L. ;,-, ,^^'t«> (2.13) From (2.2) and (2.12) the orthogonality condition S-So=0 can be expressed in the form LP + MQ + NR = 0. (2.14) The Plucker coordinates of the line [S; So] now can be written in terms of their components as [L, M, N; P, Q, R], which are known as the ray coordinates for a line (Figure 2.2). Unitized coordinates for a line can be obtained by imposing the constraint that ISI=1. The Plucker coordinates thus must satisfy equations (2.6) and (2.14) and hence only four of the six scalars L, M, N, P, Q, and R are independent. It follows that there are °°'^ lines in space^. The semi-colon is introduced to signify that the dimension of ISI is different from ISqI. Systems of lines and their properties, oo' (hne series), oo^ (congruence), oo^ (complex), are described by Hunt [13] which contains an extensive bibliography on the subject. 13 -■ V. Figure 2.2: Pliicker line coordinates A straightforward method to obtain the Pliicker coordinates was given by Grassmann (Hunt [13]) by expressing the coordinates of the points ri(xi, _yi, zi) and rjfe, J2, Zi) in the array 1 X, y^ z, 1 Xj y^ Zj and by expanding the sequence of 2x2 determinants L = 1 X, » M = 1 y. iV = 1 ^. f 1 x^ 1 ^2 1 z. y^ z, yi ^2 , Q = Z, JC, Z2 ^2 /? = ^2 y: (2.15) (2.16) 2.2 Statics of a Rigid Body The concepts developed in the previous section now can be applied directly to the statics of a rigid body. A line $ with ray coordinates [S; So] (where ISI = 1) can be used to 16 express the action of a force upon a body (Figure 2.3). Because the body is rigid, the point of application can be moved anywhere along the line. Figure 2.3: Representation of a force on a rigid body As illustrated by Figure 2.3, a force f can be expressed as a scalar multiple^ of the unit vector S that is bound to the line $. The moment of the force f about a reference point O is itio which can be written as mo = rxf where r is a vector to any point on the line $. This moment can also be expressed as a scalar multiple /So where So is the moment vector of the line $ (i.e.. So = rxS). The action of the force upon the body thus can be expressed elegantly as a scalar multiple /$ of the unit line vector, and the coordinates for the force are given by /$=/[S;So] = [f;ino] (2.17) where SS=1 and SSo=0. 17 Clearly, when the reference point O is coincident with A, then ino=0 and the coordinates of the force are [f; 0]. Therefore, f is a line bound vector that is invariant with a change of coordinate systems while mo is origin dependent. An important special case is [0; mo] which can be considered as the resultant of a pair of equal and opposite forces with coordinates [f; moi] =/[S; Soi] and [-f; moa] =/[-S; S02], where ISI=1. The coordinates of the resultant [0; ma] = [0; moi+moa] =/[0; S01+S02] are not a line bound vector, but a pure couple. The couple can be considered as equivalent to a force ^ of infinitesimal magnitude (l^-^O) acting along a line that is parallel to the lines of action of the pair of parallel forces. The line of action of 8i is infinitely distant with coordinates [0; mo], such that lpl = oo where p is the vector from the origin perpendicular to the line of action of 51, and the moment of the force <5f about the origin is px^=mo. A pure couple thus can be represented as a scalar multiple of a line at infinity. The problem of determining the resultant of an arbitrary system of forces with coordinates [fi; moi], [fj; moa], ..., [fn; mon] acting on a rigid body is essentially the determination of the quantity w=[f;mo], (2.18) where n n f=5^fi and m„=^m„,. (2.19) 1=1 .=1 It is assumed at the outset that a reference point O was chosen so that the forces acting on the rigid body were translated to point O and so that moments moi, mo2, ..., mon were introduced to yield an equivalent system of forces and torques that act on the rigid 18 body. Therefore, the line of action of the resultant force f passes through point O and the resultant moment m© is a couple [0; mo]. In general f and mo are not perpendicular (i.e., f-mo^K)). The new quantity with coordinates w = [f; mo] therefore is not a force and was defined as a dyname by Plucker. Because in general f • m© ;t 0, it is not possible to translate the line of action of force f through some point other than point O and to have the translated force produce the same net effect on the rigid body as the original dyname. The moment mo, however, can be resolved into two components, ma and mt, which are respectively parallel and perpendicular to f (Figure 2.4a) and mo = ma + mt . (2.20) The moment m, can be determined as ma = (mo -8)8 (2.21) where 8 is a unit vector in the direction of the resultant force f. The moment mt is then determined as m, = mo - ma. *^ (2.22) The line of action of force f now can be translated so that the force with coordinates [f;mt] plus the moment [0;ma] (Figure 2.4b) is equivalent to the dyname [f;mo]. Therefore, the dyname is represented uniquely by a force f acting on the line [8; 8ot] (where 8ot=-7^) and a parallel couple ma. This parallel force-couple combination was called a wrench by Ball [1]. From (2.18) and (2.20), the wrench w which is equivalent to the dyname [f; mo] can be expressed in the form 19 w = [f ; mo] = [f ; mt + ma] = [f ; mt] + [0 ; ma]. Clearly, [f; mt] is a pure force because f •mt=0. (2.23) (a) Dyname, [f; mo] (b) Wrench, [f; mt] + [0; ma] Figure 2.4: Dyname and wrench Further, because ma is parallel to f, then • ' ' ' ma = /if (2.24) where his a non-zero scalar which is called the pitch of the wrench. From (2.20) f • mo = f • ma (2.25) and from (2.24) and (2.25), the pitch h is given by f m„ f m„ h = ^ _ * '"0 f f f f (2.26) Substituting (2.24) into (2.23), together with (2.22) allows the wrench w to be written as w=[f ;mo-/?fl + [0;/zfl. (2.27) .20 ,; _:-. . ,..,^^^, Thus, the coordinates for the Hne of action of the wrench are [f; itio-Zif] and from (2.27) the equation for the hne is rxf=mo-/if. (2.28) In the same way as the action of a force can be expressed as a scalar multiple of a unit line vector, a wrench can be expressed elegantly as a scalar multiple of a unit screw $ where $ = [S;So] (2.29) and where SS= 1 . From (2.26), the pitch of the screw is given by h = SSo. (2.30) Further, from (2.2) and (2. 1 2), h = LP + MQ + NR. (2.31) Therefore, Ball [1] defined a screw as "a line with an associated pitch". Following (2.28), the Plucker coordinates for the screw axis are [S; Sq-ZiS] and the equation of the axis is r X S = So — ^S. ^2 32) 2.3 The Statics of a Parallel Manipulator Figure 2.5 illustrates a nonredundant 6-6^ in-parallel manipulator. The device has a moving platform and a fixed base connecting by six legs each of which is the same kinematic chain. The prismatic joint in each leg is actuated and the moving platform has six degrees of freedom. ■a These numbers indicate the number of connecting points in the top and base platform respectively. 21 moving platform base Figure 2.5: A 6-6 in-parallel manipulator Consider that the six leg forces with magnitudes /i,/2, ...,/6 are generated in each of the lines $i, $2, .... $6- The resultant wrench w = [f; mo] acting upon the moving platform due to these six leg forces is given by w = [fi; moi] + [f2; nioi] + . . . + [fe; mo6], (2.33) or in the alternative form, w=/i[Si; Soi] +/2[S2; S02] + ... +/6[S6; Soe] (2.34) where [Si; Soi] (ISil=l, /=1...6) are the Pliicker line coordinates of the six legs. Further, (2.34) can be expressed in the matrix form w = JF (2.35) where w = [f , hIq ]^ and F = [/, , /a , /a , A , /s , f^V are 6x1 column vector. J is a 6x6 matrix of line coordinates given as J = 22 S, Sj S3 S4 Sj Sj ^01 ^02 ^03 ^04 ^05 ^06 (2.36) and is called the Jacobian matrix, or simply Jacobian, which enables us to determine the resultant wrench w= [f; m©] produced by six actuator forces generated in the legs. It should be noted that for redundant parallel manipulators the Jacobian matrix is not square. The transpose of the Jacobian matrix relates the infinitesimal displacements 5l\ in each leg to the infinitesimal displacement twist"* of the platform and 6\ = i^6t) (2.37) where S\ = [6l^,6l2,...,5lf^f and dh = [5x,dy,5z,5(p^,6(Py,S(p^f . Here, dx, Sy, and Sz are the infinitesimal displacement of a point in the moving platform coincident with a reference point O which is chosen to be the origin of a fixed coordinate system on the fixed base. The quantities S<p^ , d(p^ , and Scp^ are infinitesimal rotations of the moving platform about the axes of the fixed reference coordinate system. In summary, the Jacobian matrix of parallel manipulators serves two distinct purposes. In its ordinary form the columns of which are the coordinates of the actuator lines (normalized), it enables us to obtain from actuated force inputs the wrench at the end effector platform. In its transposed form the Jacobian can give the relative speeds required at each actuator that corresponds to a given twist to be executed by the platform. The first of these gives the instantaneous solution to a problem of static equilibrium; the second, the solution of first order kinematic compatibility. When the Jacobian matrix is An infinitesimal twist is also a scalar multiple of a unit screw, as the scalar is an infinitesimal rotation with unit of radian. 23 singular (i.e., its rank is less than six) the actuators (i) cannot equilibrate a general wrench applied to the platform and (ii) cannot on their own prevent a transitory uncontrollable movement of the platform. This latter phenomenon is associated with the platform's gaining one or more freedoms when all the actuators are locked. The platform is then in a singularity position. CHAPTER 3 THE OPTIMUM QUALITY INDEX FOR A SPATIAL REDUNDANT 4-4 IN-PARALLEL MANIPULATOR A spatial redundant 4-4 in-parallel manipulator is shown in Figure 3.1. The device has a square platform and a square base connected by eight legs. Figure 3.2 shows a plan view of this manipulator, where the moving platform is symbolically represented by four pairs of concentric spherical joints located at points A, B, C, and D, and the fixed base is represented by another four pairs of concentric spherical joints located at points E, F, G, and H. This manipulator is said to be redundant since the platform and the base are connected by eight actuated legs. Figure 3.1: A redundant 4-4 in-parallel manipulator Figure 3.2: Plan view of the redundant 4-4 in-parallel manipulator 24 25 3.1 Determination of ^det.T^.T^ The moving platform of the redundant 4-4 parallel manipulator shown in Figure 3.2 is located at its central symmetrical configuration and is parallel to the base with a distance h. At this configuration, the manipulator is fully symmetric and each leg has the same length. Clearly, at such position the platform is most stable from the geometric static point of view. When the platform departs from this central symmetric position, the platform will lose its geometric symmetry and the eight leg lengths will be different. Therefore, it is reasonable to assume that at the central symmetric configuration shown in Figure 3.2, it is possible to determine the values of square base side b and height h based on square platform side a so that a maximum value of the square root of the determinant of the product of the manipulator Jacobian by its transpose, i.e., ^det J^J^ , may be obtained. Firstly, the coordinates of the points A, B, C, and D on the platform and E, F, G, and H on the base are determined with the origin of a fixed coordinate system placed at the center of the square base as shown in Figure 3.2, and - V2fl , B \ - — 2 2 4ia (b \ ( h , C V2a \ -- 2 2 2 2 , D H yfla h (3.1) /- -* t 2 2 A Then, using the Grassmann method described in Chapter 2 to calculate the Pliicker line coordinates of the eight leg lines, i.e., counting the 2x2 determinants of the various arrays of the joins of the pairs of points EA, FA, FB, GB, GC, HC, HD, and ED. 26 For example, the coordinates of the Hne $i are obtained using the coordinates of points E and A in (2. 15) to form the array 1 -^ 2 1 2 4la (3.2) and using (2.16) to yield S,= '42a -b , bh bh slab 1 . h\ — — , --, — : — (3.3) Similarly from points F and A, b yfla-b S,= bh _bh -Jlab 2 2 2 4 (3.4) From points F and B, S, y[2a-b b , bh bh ^lab — I — ' T' «; --:r. — —^ — : — (3.5) From points G and B, 5,= 42a -b 2' ' 2' 2' 4 (3.6) From points G and C, _^ -jla-b bh _bh yf2ab 2' 2 ' ' 2' 2' ^~ (3.7) From points H and C, ^6 = ^ V2a-j? K ^ bh Sab 2' 2 ' ' 2' 2' 4~ (3.8) From points H and D, 27 S,^ Sa-b _b_ , bh bh ^Jlab 2 2 2 2 4 (3.9) From points E and D, 5„ = yfla-b b 1 bh bh 2' 2 h; — — , — -, - 42ab (3.10) It should be noted that the above Pliicker line coordinates are not normalized and each line must be divided by /, = |Si| (/ = 1, 2, ..., 8). Hence, the normalized Jacobian matrix of the eight leg lines (now all reduced to unit length) can be expressed as J = si si Si si si si si si L^> h h h h h I, h (3.11) Since the device is in a synunetrical position, the normalization divisor is the same for each leg, namely /, = /(/= 1,2, . . ., 8), and for every leg l = ^L^+M^+N' =J-(a'-yf2ab + b' + 2h'). (3.12) From (3.3) to (3.10), the Jacobian matrix in (3.11) becomes J=- / b 2 h bh 2 bh 2 y[2ab _b 2 -d, b 2 bh _bh 2 2 _bh _bh 2 2 ■\j2ab 42ab 2 b_ 2 d, h bh bh 2 2 _bh _bh 2 2 s2ab 42ab 2 -d, 2 bh bh 2 2 bh bh 2 2 \2ab \2ab -d, b 2 h bh 2 bh 2 ^ab (3.13) 28 where d,= 42a-b Using equation (3.13), the determinant of the product J J ^ can be expressed in the form detJJ^=- / 12 ^2 -d. d. d. 8/1^ d. Ib'h' -d. 2b'h' aW where d^=2{a'-42ab+b''), d^^{42a-2b)bh. Expanding (3.14) and using (3.12), then extracting the square root yields (a^-^l2ab + b^+2h^} (3.14) (3.15) Assuming the top platform size a is given, now taking the partial derivative of (3.15) with respect to h and b respectively and equating to zero yield and 9642a'b'h'(a'-yf2ab + b^-2h') (a^-yf2ab + b^+2h^} 96Sa'b'h'{a'-b'+2h') _^ [a^ -42ab + b'' +2h^)' (3.16) (3.17) 29 When a, b, and h are not equal to zero, equations (3. 16) and (3.17) give a^-^Jiab + b^-lh' =Q, (3.18) a^-b^+2h'=0. (3.19) Adding (3.18) and (3.19) yields 2a^-42ab = 0. (3.20) Solving the above equation, we obtain b = 42a. (3.21) Further, substituting (3.21) into equation (3.19) yields a^-2h^=Q. (3.22) There are two solutions for h in the above equation, here we only take the positive solution (the negative solution is simply a reflection through the base) " = ;!• ... . ;.., . (3.23) Finally, substituting (3.21) and (3.23) into (3.15) we get VdetJ^J: = {VdetJJ^)^ = A^la' (3.24) where Jm denotes the Jacobian matrix for the configuration at which the 4-4 redundant parallel manipulator has a maximum quality index. This optimum configuration is shown in Figure 3.3. 30 Figure 3.3: Plan view of the optimal configuration of the redundant 4-4 in-parallel manipulator with the maximum quality index 3.2 Implementation From the definition of quality index (see (1.2)) and (3.24), the quality index for the redundant 4-4 parallel manipulator shown in Figure 3.1 becomes VdetjJT Z = - 4yf2a^ (3.25) The variation of the quality index now is investigated for a number of simple motions of the top platform. Here, an optimal redundant 4-4 parallel manipulator with platform side a = 1 , and thus base side Z> = V2 is taken as an example. First, consider a pure vertical translation of the platform from the central symmetric position shown in Figure 3.2 along the z-axis while remaining parallel to the base. For such movement, from (3.15) and (3.25), the quality index is given by Sb'h 31 31.3 {a^-yl2ab + b^+2h^J With a = 1 and ^ = v2 , this reduces to (3.26) A = \e42h^ (3.27) and is plotted in Figure 3.4 as a function of h. It shows that at height h = -p = , the -v2 2 quality index of the redundant 4-4 parallel manipulator has a maximum value, A = 1 . 1.0 - 0.8 - ■S 0.6 - c 3 0.4- o 0.2- 0.0- f\ \ \ 1 \t^ i ■ ■ t ■■ 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Height h Figure 3.4: Quality index for platform vertical movement We now derive an expression for -^det JJ^ when the platform of the redundant 4-4 parallel manipulator is translated away from its central location while remaining parallel to the base at height h. Assume the center of the moving platform to move to point (x, y, h), then the coordinates of the points A, B, C, and D on the platform become 32 X y- 42a X y-\ h \ B D x + - 42a X-- 42a y h y h (3.28) The coordinates of points E, F, G, and H on the base can be found from (3.1). Thus, the Pliicker line coordinates for each of the eight leg lines can be determined as (3.29) (3.30) s^ = b 42a-b ;"2' '- 2 ' h; bh 2' bh 2' b{2x-2y + 42a) 4 s,= b 42a-b ['-2' '- 2 ' h; bh 2' bh 2 b{2x + 2y-42a) 4 42a-b b h; bh 2' bh 2 b{2x + 2y + 42a) 4 42a-b b 2-^2 h; bh 2' bh 2' b{2x-2y + 42a) 4 Ss = b 42a-b ^2' '^ 2 ' h; bh 2' bh 2' b{2x-2y-42a) 4 b 42a-b ^2' '^ 2 ' h; bh 2' bh 2' b{2x + 2y + 42a) 4 J' 42a-b b r 2 ' '-2' h; bh 2' bh 2' b{2x + 2y-42a) 4 J' Ss- 42a-b b r 2 ' '^2' h; bh 2' bh 2" b{2x-2y-42a) 4 (3.31) (3.32) (3.33) (3.34) (3.35) (3.36) 33 The above coordinates are not normalized and each row must be divided by the corresponding leg length. The Jacobian matrix J then can be constructed by using (3.11). Further, substituting and expanding -^det JJ^ yields (3.37) where the leg lengths are h=- li=- ( ^1 2 / X + I 2j V y+- -^a-b + h' 42a-b"\ ( M' X — - r-h +h' l2=- h=- h = /«=. ( b^ X — V 2, + Sa-b + h\ 42a-b^ ( b^' JC + - + r-^, +h' ( ^Y ( 4la-b \2 X + V 2, y + - + h\ ^ 4ia-b^ ( b^' X-- J* 2, + h' ^ With a = \,b = yf2,mdh = , from (3.25) and (3.37), the quality index A becomes VdetJJ^ ^d^iJJl ~ ixU2x'y'+y' +2x'+l){xU2x'y'+y* +2y'+\) ' x^+y^+] (3.38) which is plotted in Figure 3.5(a) as a function of x and y. Figure 3.5(b) shows the contours of the quality index for this platform horizontal movement. The contours are labeled with values of constant quality index and they are close to being concentric circles of various radii. When j: or _y is infinite, A=0, and when x=y=0,X=\. 34 1 -1 -0.5 - -1.0 (a) (b) Figure 3.5: Quality index for platform horizontal translation To illustrate the variation of the quality index during some simple rotations of the platform, a new coordinate system x'y'z' is attached to the square platform. The origin of the new coordinate system is located at the center of the top platform and the coordinate system is oriented such that its a; '-axis is passing through vertex B, its ^''-axis then is passing vertex C, and its z'-axis is normal to the square platform ABCD. Thus, when the platform locates at its initial central position shown in Figure 3.2, the x'- and ^''-axes are parallel to the x- and >'-axes on the base respectively. Figure 3.6 illustrates a side view of the redundant 4-4 parallel manipulator when the square platform ABCD is rotated by an angle dy about the y'-axis from its initial position. For such platform rotation, the coordinates of the vertices A, B, C, and D become 35 Sa yfla \ f h B J \ \ ( » D / v cos^„ h + sin^„ 2 ^ 2 ^ ^[2a 42a cos9^. h- 2 ' 2 sin 0. (3.39) Figure 3.6: Platform rotations about the >> '-axis It should be noted that the positions of line $i, $2, $5, and $6 do not change during this movement and their corresponding Plucker line coordinates can be obtained from (3.3), (3.4), (3.7), and (3.8) respectively. The Plucker coordinates for the line $3, $4, $7, and $8 are now given by ^f2a cos 0^. -b b 42a sin 0^. + 2h S,= 2 ' 2' 2 ' b(42a sin 0^ + 2h) b{42a sin 0^ + 2h) 42ab cos 0^. (3.40) 36 ^4 = ■\J2acosdy-b b yflasinO^ +2h 2~ ' "2' 2 ' bi-yfla sin 6^ + 2h) b{42a sin d^ + 2h) y/lab cos 9^ 4 (3.41) S, yfla cos ^^ - ft ft 42a sin ^^ - 2/i 2~ ' "2' 2" ' b(^I2a sin ^^. - 2/i) b{^|2a sin ^ - 2/z) ^f2ab cos 6' (3.42) 5« = ^J2a cos ^, - ft ^ V2a sin ^^ - 2/i 2 2 2 Z7(V2a sin ^^, - 2h) b{42a sin ^^, - 2h) -Jiab cos ^^ (3.43) Since the configuration of the manipulator keeps symmetric about the x-axis during the platform rotation about the j'-axis, from Figure 3.6, we have /, = /2 = /j = /g = / , /j = /4 , and /, = /g . The corresponding Jacobian matrix then can be determined by (3.1 1), and further ^det JJ^ becomes 4hi-y/2a-b){l^-l^)sme^+2i4h^cos^e^+b^sm^e^)l^) {h\4h^cos'e^ +b^sm^e^)(l^ +l^)-Abh\ll -l^)cose^.^md^ + (3.44) ((4a'/z' cos' e^ +a^b^ sin' ^^ + 2b^h^ -^^abh" cos^Jsin' ^^ + 8/i'cos^ )/'))2 where 37 I, =-^{j2acose^ -bf +b^ +{y[2asme^ +2h)\ Ij = -^(V2fl cos d^-bf+b^+ (Jla sin 6^ -2h)\ and / can be found from (3.12). S With a = \, b = ^f2, and h = — , from (3.25) and (3.44), the quaUty index becomes A = -y/detJJ^ _ -^(3 - cos 6' J(6 cos ' 0^. - cos ' 6^. - 7 cos ^,, + 4) VdetJ™J; ~ 2(2cos'^^, -4cos^,-l-3) (3.45) Since the redundant 4-4 in-parallel manipulator is fully symmetric at its central configuration shown in Figure 3.2, the same result can be obtained when the platform is rotated about the jc'-axis. From (3.45), the variation of the quality index for rotations about the x'- and y'- axes is drawn in Figure 3.7. Rotation Angle (degree) Figure 3.7: Quality index for platform rotations about the x'- and y'-axes 38 Figure 3.8 illustrates a plan view of the redundant 4-4 parallel manipulator with the moving platform ABCD rotated ^ about the z-axis. The x and y coordinates of the vertices A, B, C, and D then become x^=rsm6^, y^=-rcos6^, Xg=rcosd^, yg=rsin9^, X(^ =~rs'md^, Jc =rcos6^, Xu =-rcosd^, yD=-rsin6^ (3.46) a where r = —^ . From (3.46) V2 x^+Xg+Xc+Xj,=0, (3.47) The complete set of coordinates of points A, B, C, and D are therefore A{x^ y^ h), B{x, y, h) , C{xc yc h) . D{x^ y^ h) (3.48) where h is the height of the moving square ABCD above the base square EFGH. -(v- -. ^ __ ^§^ ex -t) 1 m __^ ^"-^ u 1^ III * ' \ h \ /' ^T? .J! rl' / /A^^ a: iTvJw \ ^ \ Figure 3.8: Platform rotations about the z-axis 39 Then the coordinates for the corresponding lines $i, $2, . . ., $8 are given by 5,= b b , bh bh b{x^-yj "2^2 22 2 (3.49) S,= Xa — >'a+T' ^' -^T' - bh bh bix^ + y^) (3.50) S3 = b b bh bh b(Xg + yg) (3.51) 5,= ■^5 = ^B 2' ^« -^c --' >'c 6 /i; bh fo/i 2' 2' 2 2' /i; 2' 2 ^Ub->'b) K^c-^c) (3.52) (3.53) ^6 = b b , bh bh b(Xr + Vr ) ^ 2 "^ 2 2 2 2 (3.54) 5,= V 4-^ V -.^ h- ^ ^ Hxp + yp) '2' ^^ 2' ' 2' 2' 2 (3.55) S,= b b , bh bh b{Xry-yn) "2' ^"2' ' 2' 2' 2 (3.56) It is apparent from Figure 3.8 that 1^=1^=1^=. l^ and 1^=1^=1^=1^. The corresponding Jacobian matrix can be determined by (3.11). Furthermore, calculating ydetJJ^ yields VdetJJ^ = 4V2a'ft'/i^|cos/9, (3.57) where 40 /,=. l2 = 42a sin 0+b V r 42acose, -b^ + h\ 42asmd,-b^ (42acos0,-b ,, + h With a-l,b = V2, and h = — , from (3.25) and (3.57), the quality index becomes X = cos 0, jy/detJJ^^ V'l^t J m J m (2 COS^ ^^ - 4 COS ^^ + 3)2 (3.58) This is plotted in Figure 3.9 and it shows how the quality index varies as the platform is rotated about the vertical z-axis through its center. The eight legs are adjusted in length to keep the platform parallel to the base at a distance h. It is shown in the figure that the manipulator has the highest quality index A = 1 when ft = 0°, and A = (singularity) whenft=±90°. 1.0- 1 0.6- .^- 3 0.4- O 0.2 - 0.0- /'^\ K y 1 i i i ^ 30 60 90 Rotation Angle (degree) Figure 3.9: Quality index for platform rotation about the z-axis 41 It is interesting to note that the redundant 4-4 parallel manipulator shown in Figure 3.1 always becomes singular when its platform rotates ^^=±90° about the z-axis from its central symmetric position. This can be seen from (3.57), -^det JJ^ = when dz = ± 90°. Figure 3.10 illustrates the singularity position of the redundant 4-4 parallel manipulator when 6z = 90°. It is not immediately obvious from the figure why the eight connecting legs are in a singularity position. This kind of singularity has been discussed in detail by Hunt and McAree [14]. They explain that at such position, even when all eight leg actuators are locked, the connectivity between the base and moving platform is one. The moving platform can move instantaneously on a screw reciprocal^ to the eight leg forces on the z-axis with pitch h^, i.e., a screw with coordinates [O, 0, 1; 0, 0, h]. (3.59) H y{ I G V E ^ SK \ --♦♦y^ \ \ X F Figure 3.10: Plan view of the singularity position of the redundant 4-4 in-parallel manipulator when 6^ = 90° When a wrench acts on a rigid body in such a way that it produces no work while the body is undergoing an infinitesimal twist, the two screws are said to be reciprocal. 42 Now when ^z=±90°, from (3.49) through (3.56), the component of moments about the z-axis for each of the eight legs are all the same -• A^.. = h and R, = ± ■Jlab (/ = 1,2,...,8) (3.60) The coordinates for the eight legs become 5.= L,, M,, h; P^, e,, ± -Jlab Hence, from (3.59) and (3.61), h,h± = or h,=+ (3.61) (3.62) 4 ' Ah It follows that all eight legs lie on a linear complex, which is a three-parameter system of linearly dependent lines (Hunt [13]). CHAPTER 4 THE KINEMATIC ANALYSIS OF THE SPATIAL REDUNDANT 4-4 IN-PARALLEL MANIPULATOR The kinematic analysis of in-parallel manipulators deals with the study of the platform motion determined by the leg displacements. Two problems can be distinguished for the kinematic aspects: inverse kinematics and forward kinematics. The inverse kinematics problem, i.e., finding the leg lengths for a given location (position and orientation) of the mobile platform (a difficult problem for serial manipulators), is straightforward for parallel manipulators. On the other hand, the forward kinematics problem, i.e., finding the platform location from a given set of leg lengths, is much more difficult. In general, this problem has more than one solufion for nonredundant cases. As an example, the forward analysis for the general 6-6 platform requires the solution of a 40* degree polynomial (Raghavan [31]), the solution of which is clearly impractical for real-time implementation. A lot of methods have been presented to solve various types of nonredundant parallel manipulators as summarized by Dasgupta and Mruthyunjaya [7]. However, few works have been done on the kinematic analysis of redundant parallel manipulators, particularly the forward kinematics. A similar problem for determining a unique position and orientation of the platform of a general geometry parallel manipulator by using seven linear transducers has been solved by Innocenti [15]. He constructed a 146x147 constant matrix for solving the problem, which makes the computation time still larger than real time. Also, Innocenti's method produces only one solution for a general geometry parallel manipulator, but when the fixed base of a 43 44 manipulator is in a plane, there should be two solutions instead of one since the manipulator can have two reflection configurations through the base according to the same set of readings from linear transducers no matter how many transducers are used. Therefore, his method may not be correct to solve the problem when the base of a parallel manipulator is in a plane. Zhang, Crane and Duffy [37] have performed the forward kinematic analysis on a special redundant parallel mechanism whose platform and base are connected by a ball- and-socket joint with four legs to determine the orientation of the platform. In this chapter the kinematic analysis of the redundant 4-4 in-parallel manipulator shown in Figure 3.1 is performed. From here on, the notations from Crane and Duffy [5] are used to describe the coordinates of point and the transformation matrix. For example, the notation Pa is used to indicate the coordinates of a point A as measured in a coordinate system I and ^R is defined as the orientation of the coordinate system 2 relative to the coordinate system 1 . 4. 1 Inverse Kinematic Analysis The objective of the inverse kinematic analysis of the redundant 4-4 parallel manipulator is to find the eight leg lengths for a given position and orientation of the square moving platform. In Figure 4.1, coordinate systems 1 and 2 have been attached to the fixed base and the moving platform respectively. The origins of the coordinate systems 1 and 2 are located at points £ and A receptively. The coordinates of points E, F, G, and H on the base are known in terms of the coordinate system 1 and 45 b b Pe = 'p - 'P - b 'P = * H b (4.1) while the coordinates of points A, B, C, and D on the platform are known in terms of the coordinate system 2 and (4.2) a a Pa = ^p - ^P - * c a ^P - a Figure 4. 1 : Coordinate systems of a redundant 4-4 in-parallel manipulator 46 It is well known that the location of a rigid body in space can be described by the position and orientation of a coordinate system attached to the body with respect to a fixed reference frame. Thus, for the inverse analysis, the position and orientation of the coordinate system 2 is considered known, and can be given by the position vector ' P^ , which defines the position of the origin A of the system 2 relative to the origin E of the fixed frame 1, and a rotation matrix jR, which is a 3x3 matrix whose columns are the unit vectors along the coordinate axes of the system 2 as measured in the system 1 . Then the coordinates of points B, C, and D in terms of the coordinate system 1 become Pc - Pa + 2^ Pc 'P, = 'P, + ]R^P,. Finally, the eight leg lengths can be calculated by =V('Pa-'Pe)-('Pa-'Pe). =V('Pa-'Pp)-('Pa-'Pf). = V('Pb-'Pf)-('Pb-'Pf). =V('Pb-'Pg)-('Pb-'Pg). =V('Pe-'Po)-('Pc-'PG). =>/('Pc-'Ph)-('Pc-'Ph). 7=V('Pd-'Ph)-('Pd-'Ph). =V('Pd-'Pe)-('Pd-'Pe). (4.3) (4.4) where ■ represents the dot product of two vectors. 47 Hence, for a given location of the moving platform, there is only one possible solution for each leg length. 4.2 Forward Kinematic Analysis The objective of the forward kinematic analysis of the redundant 4-4 parallel manipulator is to find the location of the moving platform given the actuator displacements of all the eight legs. Thus, the coordinates of points A, B, C, and D measured in the coordinate system 1 shown in Figure 4.1 need to be determined for a given platform side a, base side b, and eight leg lengths /, (/= 1, 2, ..., 8). 4.2.1 Introduction The forward analysis is performed in detail in this section and thus provides a unique solution for the location of the moving platform above the base platform together with a reflected solution through the base for an arbitrarily specified set of eight leg lengths. However, extreme care must be taken in applying this analysis since what appears to be an arbitrary set of leg lengths may well be special and the solution will fail. For such cases, the constraint equations employed in the analysis presented here become linearly dependent in one way or another. A class of special cases has been reported by Selfridge [32] where he obtained a pair of assembly configurations (as opposed to a unique solution) above the base and a corresponding pair of reflected solutions through the base. It is interesting to note that one class of solutions reported by Selfridge [32] occurs when the platforms are parallel, the odd leg lengths are all equal and /, = /j = /, = /^ = / . Further, the even leg lengths are all equal and 1^=1^=1^=1^=1'. This class of solution embraces the workspace generated by a rotation of the top platform about the z-axis (Figure 3.8). While this does 48 not raise a problem with the quality index analysis it is important to recognize that these are in fact a pair of assembly configurations above the base platform. A numerical example is present in section 4.2.5. All this of course raises the issue of other classes of special cases that are worth further investigation. 4.2.2 Coordinate Transformations First, the coordinates of any point in coordinate system 2 need to be transferred to coordinate system 1 . To do so, coordinate system 2 may be obtained by initially aligning it with coordinate system 1 and then introducing the following transformations: 1 . Rotate the coordinate system 2 by an angle d\ about the ;c-axis until the >'-axis is in the plane defined by points A, E, and F, and the scalar product of the >'-axis with the vector Si is positive as shown in Figure 4.2. 2. Translate the origin from point Eto F along the positive x-axis. 3. Rotate by an angle ^ about its current z-axis, which causes the jc-axis to point along the vector S2. 4. Translate the origin from F to A along the negative x-axis. 5. Rotate the coordinate system about its current jc-axis by an angle 62 until the y-axis is in the plane defined by points A, F, and B, and the scalar product of the ^'-axis with the vector S3 is positive. 6. Rotate by an angle ^ about its current z-axis, which causes the a:-axis to point along the vector S3. 7. Rotate by an angle ^ about its current jr-axis until the y-axis points along the vector S4. 49 Figure 4.2: Coordinate transformations The coordinates of points A, B, C, and D may now be expressed in the coordinate system 1 as ?. ' 'P3 = R,K+R3(t,+R,R,R/P3)], 'Pc = r,[t,+R3(t,+r,r,r/pJ], 'p, = r,[t,+R3(t,+R3R,r/pJ], where (4.5) 50 1 b COS0, -sin0, 0" '-h R,= cos^, -sin^, . T2 = » R3 = sin0, COS0, T = sin^, cos^, 1 1 - cos 02 -sin 02 '1 R5 = cos ^2 -sin ^2 . R6 = sin 02 cos 02 . R7 = COS ^3 -sin ^3 sin ^2 cos ^2 1 sin ^3 COS ^3 and the coordinates of points A, B, C, and D are known in terms of the coordinate system 2 and are written as ^P^, ^Pg, ^P^, and ^P^. The angles 0i and 02 are shown in Figure 4.2 as the inner angles of the triangles AEF and AFB, respectively. Therefore, the angles 0i and 0^ are constrained to lie in the range of to K. The cosines of 0i and 02 may be determined from a planar cosine law as cos 01 = cos 02 b'+ll-l^ 2bU a^+ll-ll (4.6) laL and the values of 0i and 02 are determined as the inverse cosine value in the range of to n. The coordinates of points A, B, C, and D as measured in the coordinate system 1 have been written as a function of the parameters di, 02, and ft. The objective now is to determine these parameters that will locate points A, B, C, and D such that they satisfy the distance constraints with points E, F, G, and H. 51 4.2.3 Constraint Equations Since the eight leg lengths, /, (/=1, 2, ..., 8), have been given for the forward analysis, the distance between points A, B, C, and D on the platform and points E, F, G, and H on the base must satisfy these leg lengths as shown in Figure 4.2. The coordinates of points A, B, C, D, E, F, G, and H all have been expressed in terms of the coordinate system 1 (see (4.5) and (4.1)), and the distance between these points may be expressed in the coordinate system 1 as ('P3-'p^)■('P3-'p,)=/^ : (4.7) ('P,-'P^)-('Pc-'Po) = /5\ (4.8) ('Pc-'Ph)-('Pc-'Ph) = ^6. (4.9) (•P„-'Ph)-('P^-'Ph) = /^ (4.10) ('P^-'P,)•('P„-•p,) = /^ (4.11) Note that three constraint equations are not written for the distance between points A and E, A and F, and B and F. The distance between these points will be equal to /i, I2, and I3, respectively. These three leg lengths have been used in the transformation of coordinate systems, which relates the coordinate systems 1 and 2 included rotation angles 0i and ^ and origin translation distance Ij. Equations (4.7) through (4.1 1) may be expanded and factored into the form labs^SiS^ -2bcj(as^c^c-^+s^(ac^ -l2))+a^ +b^ -^ac^l^ +ll - l] =0, (4.12) labs^ (s^ s^ + c^ s^Cj + c^s^ ) + Ibc, [ac^ s^s^ - ac^ s^c^ - a{c^ c^ s^ - s^ s^ )c^ - s^{ac^ -l2))+2a^+b^ -lac^l^ +ll -ij =0, (4.13) 52 s^ iac^ - ^2 ))+ 2aZ75^ (^2^3 -s^c^)- 2a(bs^ c^ c^ + (bc^ - 1^ )s^ )c^ + (4. 14) 2a^ +2b'^ +2abc^c^ -2bc^l-^-2ac^l^+ll -ll =0, 2abs^ (c^ s^c^ + c^s^ ) + 2bc^ [ac^ s^s^ - a(c^ c^^ c^ - s^ s^^ ^-^ + l^s^ )+ 2abs^ s^s^ - 2a[bs^ c^ c^ + {bc^ - 1^ )s^ )c^+a +2b - 2bc^ l^+l^-lj =0, 2abs^s^s^-2a{bs^c^c^ +bc^s^ -IjS^^Y^ +a^ +b^-2bc^l^ +ll -1^=0 (4.16) after recognizing that sf +cf =1 and s^ +c^ =1, and where Si, c„ (/= 1, 2, 3) represent the sine and cosine of di and s^ , c^ ,(j= 1, 2) represent the sine and cosine of ^. The objective now is to determine values for 61, &i, and ft which will simultaneously satisfy the five equations represented by (4. 1 2) through (4. 16). 4.2.4 The Solution To solve equations (4.12) - (4.16), we consider s\, c\, S2, c^, 53, and C3 as independent variables. It gives us three more equations since sin ' d, + sin ' 6, = sf + cf = I, {i = 1, 2, 3). (4.17) Now equations (4.12) - (4.16) will be manipulated to eliminate ^1, c\, 53, and C3 first. The algebra to achieve this is what follows. Adding equations (4.12) and (4.15), and then subtracting (4.14) yields k^c^+k^C2+k^=0, (4.18) where ki, k2, and ^3 are known constants and are defined in Appendix A. Similarly, an equation that is linear in cj and C3 is generated by subtracting (4.14) from the sum of (4.13) and (4.16): k2C^+k^Cj+k^=0. (4.19) 53 Solving equations (4.18) and (4.19) for c\ and c^ respectively yields c. =-r-c2-r-. (4.20) ^3 ~ ; ^2 ~ (4.21) k k Now substituting the expressions for c\ and C3 into equations (4.12) and (4.16) produces kf^s^S2 + k^c\ + /CgCj + /:, =0, (4.22) k^s^s-^ +k^Qsl+k^^c^^ +/:,2 =0. (4.23) Solving the above two equations for s\ and .^3 respectively yields ^^^_ V2+V2+^9 ^ (4.24) "^6-^2 „ _ ^10-^2 "^^11^2 "*""^12 /A^c\ Finally, substituting the expressions for ^i, 53, c\, and cj, into equations (4.15) and 5,. + c,. =1 for i = 1 and 3 produces three equations in two variables, 52 and C2. Further, replacing ^2 with 1 - Cj , it is interesting to note that S2 cancels from these equations. This leaves following three equations in only one unknown, C2 Eq, (C2 ) = X M,c^' = (/ = 1, 2, 3) (4.26) where the constants My are defined in Appendix A. The objective now is to determine value for C2 that simultaneously satisfy the three equations represented by (4.26). 54 Multiplying the three equations in (4.26) by C2, we obtain three additional equations. Thus, a total of six equations in the unknown C2 are obtained. These equations can be written in matrix form as My=0 (4.27) where M= M,4 Af,3 M,2 M,, M,o ^24 ^23 ^22 ^1\ ^2(i ^34 -^33 ^32 ^31 ^30 ^ Af,4 M,3 M,2 M,, M,o M24 M23 M22 M21 Mjo M34 M33 M32 M3, M30J y = Here, we treat Cj , c^, Cj, Cj, Cj, and 1 as unknowns and thus equation (4.27) can be regarded as a homogeneous linear system in six unknowns. The trivial solution of y = is not feasible, since the last element of y must equal 1 . Solutions other than the trivial solution exist only if the homogeneous equations are linearly dependent, and as such the determinant of the matrix M must equal zero. Evaluating this determinant and seeing how close it is to zero will provide an indication of the quality of the measured data (i.e., the platform side a, base side b, and the joint positions) and the sensed data (i.e., the eight measured displacements /], /i, ..., k). The issue of how close to zero is satisfactory is not addressed in this dissertation. The six equations represented by (4.27) may now be rearranged into the form Ux = V (4.28) where 55 U = M,4 M,3 M,2 M,, M,o ^24 ^23 ^^22 ^2\ ^20 M34 M33 M32 M3, M30 M„ M,3 M„ M„ Af24 M23 M22 M21 M34 M33 Af32 M3, J ■-2^" -2^ x = c,^ v = > C2^ -A^.o -M20 ^2] _-M3o_ Equation (4.28) represents six linear equations in five unknowns. The vector x may be solved for by selecting any five of those equations. The term cj is the fifth component of the vector x and unique value for this term is thereby determined. However, it should be noted that since &i is in the range of to In, there are actually two solutions of Si for a value of ci. Thus, the manipulator has two configurations for a given set of leg lengths. These two configurations are due to a reflection through the base plane. For each value of &i, corresponding values for c\, cs, s\, and 53 can be calculated from (4.20), (4.21), (4.24), and (4.25) respecfively. Then, values for 0i and <% can be determined. Finally, the coordinates of points A, B, C, and D in terms of the coordinate system 1 can be obtained by substituting 6[, 62 and 03 into (4.5). 4.2.5 Numerical Verification In this section, a numerical example is presented for a redundant 4-4 parallel manipulator to verify the analysis. The dimensions of the manipulator are measured in an arbitrary length unit and given as follows: platform side a = 10, base side b=l5. A set of leg lengths are given as 56 /, =13.62421, /^ =10.40411, /j =14.47201, 7^=1 1.16409, /, =16.34095, /^ =17.59696, /^ =16.22984, /g =15.92500. The numerical results are presented in Table 4. 1 and two configurations are shown to be reflecting through the base plane. Thus, a unique configuration may be easily determined by checking the sign of z coordinate of one of the platform joints. In order to verify these results, an inverse kinematic analysis was performed. All solutions reproduced the correct leg lengths. Table 4. 1 : Numerical results of the redundant 4-4 in-parallel manipulator No. ^1 (deg.) Oi (deg.) Si (deg.) Pa 'P 'Pc ■p„ 1 -105.534 133.523 -27.872 "10.079" 2.455 8.832 "16.119" 10.327 10.077 " 8.921" 15.045 15.168 " 2.881] 7.173 _13.923j 2 105.534 -133.523 27.872 BSS '10.079" 2.455 -8.832 " 16.119" 10.327 -10.077 8.921' 15.045 -15.168 2.881] 7.173 -13.923] In the following example the above solution failed because equation (4.7) through (4.11) become linearly dependent while equation (4.9) through (4.11) are redundant for the system. Using the same dimensions as the above example, the leg lengths now become: ^1=^3=^5=^7=^ = 18 and l^=l^=l^=l^=r=\e. 57 The numerical results are presented in Table 4.2. It is apparent that there are now two configurations above the base plane and further two solutions reflected through the base. Table 4.2: A numerical example for the special case of the redundant 4-4 in-parallel manipulator No. h K P 'Pc P '2.267" '-6.698' "-2.267" " 6.698" 1 5.199 6.698 5.199 2.267 5.199_ -6.698 . 5199 -2.267 5.199 ■ 2.267' ' 6.698' "-2.267" "-6.698" 2 15.099 -6.698 15.099 2.267 15.099 6.698 15.099 - 2.267 15.099 " 2.267' '-6.698' "-2.267" " 6.698" 3 -5.199 6.698 -5.199 2.267 -5.199 -6.698 -5.199 -2.267 -5.199 r 2.267 ■ 6.698" "-2.267" -6.698" 4 -15.099 -6.698 2.267 6.698 -2.267 ■ -15.099 -15.099 -15.099 -15.099 CHAPTER 5 THE OPTIMUM QUALITY INDEX FOR A SPATIAL REDUNDANT 4-8 IN-PARALLEL MANIPULATOR A redundant 4-8 parallel manipulator is shown in Figure 5.1, which is derived simply by separating the double ball-and-socket joints in the base of the redundant 4-4 manipulator shown in Figure 3.1. A plan view of the redundant 4-8 parallel manipulator is shown in Figure 5.2. The device has a square platform of side a and an octagonal base formed by 4 pairs of joints Ed and Ea, Fa and Fb, Gb and Gc, and He and Hq. Each pair of joints is separated from a vertex of a square of side t by a distance fib for which 0</3<— . Clearly the platform is degenerate when fi= —. Figure 5. 1 : A redundant 4-8 in-parallel Figure 5.2: Plan view of the redundant 4-8 manipulator in-parallel manipulator 58 59 5.1 Determination of ^det J^ J The moving platform of the redundant 4-8 parallel manipulator shown in Figure 5.2 is located at its central symmetrical configuration and is parallel to the base with a distance h. By analogy with the redundant 4-4 parallel manipulator, a maximum value of the square root of the determinant of the product of the manipulator Jacobian by its transpose, i.e., ■^'^et J^jj^ , may be obtained for this configuration. A fixed coordinate system is placed at the center of the octagonal base as shown in Figure 5.2. Then, the coordinates of the points A, B, C, and D on the platform are -^ h , B } 4ia h \ ( , C ^ h , D 42a h (5.1) The coordinates of the points Ea, Fa, Fb, Gb, Gc, He, Ho, and Ed on the base are r Gc ( -^4 -- ^ 2 . F, d, -- ■d, V . G, \ ^ J \ ( , Hf, , Hu d, (h ^ d, \2 ; h ^ ~2 -^4 C ) J (5.2) where ^4 = {l-2j3)b Counting the 2x2 determinants of the various arrays of the joins of the pairs of points EaA, FaA, .... EdD yields the Pliicker line coordinates of the eight leg lines. That is, from points Ea and A, 5,= , V2a-ft , bh ^ , Sad. d„ — , h; - — , d,h, —^ (5.3) From points Fa and A, 60 S,= , -Jla-b , bh , , -Jlad^ -d„ — , h; -—, -d^h, ^ (5.4) From points Fb and B, Sa-b S,= ^ u J u ^^ 42ad^ , d^, h; -d^h, — — , — -— 2 2 2 (5.5) From points Gb and B, \[ia-b S,= , , , , bh -Jlad, , -J4, h; d^h, — — , -— 2 2 2 (5.6) From points Gc and C, ^5 = -d,. 4ia-b , bh -d^h. 42ad^ (5.7) From points He and C, k = , yfla-b , bh , . ^l2ad^ d„ ^ ' ^' 2 ' ' ^ ■ From points Ho and D, S,= 42a-b ^ , ^ , bh 42ad 2 ' ^4. h; d,h, ^ , ^ i_ (5.8) (5.9) From points Ep and D, S,= 42a-b , J4, h; -d^h. bh slad^ (5.10) The normalized Jacobian matrix of the eight leg lines can be expressed in the form J = \sj si si si si •^6 si ['■ h h U h h l. k (5.11) 61 where h, k, .■.,k are the leg lengths and since the device is in a symmetrical position, I] = l2 = ••■ = /g = /, and l = ^|L^+M^+N^ =J-[a^ -42ab + {2j3^ -2j3 + l)b^ +2h^]. (5.12) From (5.3) to (5.10), (5.11) becomes d^ -d^ d^ di -d^ d^ -dr -d, -d, -d, d. -d. d, d, -d. h h h h h h h -7 bh 2 bh 2 -d^h d^h bh 2 bh 2 d^h d,h -d,h bh 2 bh 2 -d,h d,h bh 2 42ad, ■sliad. 42ad, 42ad, Sad, Sad, Sad, -d,h bh 2 Sad, (5.13) where same as (3.13), d^ = Sa-b A{dl+dl) detJJ' = From equation (5.13), the determinant of the product J J becomes 2{2dl-bd,)h 4(rf'+^4) 2{bd,-2dl)h 8/i' 2{bd,-2dl)h (b^+4dl)h^ 2i2dl-bd,)h (b^+4d^)h^ T 1 /'^ 4a' d 2 J2 (5.14) Expanding (5.14) and using (5.12), then extracting the square root, we obtain 62 VdetJJ =-r — ^ yr. (5 15) y -ynab + {2fi^-2p + l)b'+2h'\ Assuming the top platform size a is given and taking the partial derivative of (5.15) with respect to h and b respectively and equating to zero yield 9642{l-2/3ya^b^h^[a^-yf2ab + (2fi^-2fi + \)b^-2hA ^* r 7^ T4 ~^ (5.16) [a^-yl2ab + {2fi^-2fi + l)b^+2h^\ and ' ^ * ' I ^ 96yf2{-\ + 2fi)'a'b'h'[-a' +(2fi' -2/3 + \)b' -2h'] _^ \ t— i4 ~ (5-17) y-42ab + {2P^-2p + \)b^+2h^ Note that we already assumed fi^ — , then when a, b, and h are not equal to zero, from equations (5.16) and (5.17), we get a'-42ab + {2p^-2p + \)b^-2h^ =0, (5.18) -a^+{2p^-2p-\-\)b^-2h' =Q. (5.19) Subtracting (5.19) from (5.18) yields 2a^-y[2ab = 0, (5.20) and thus b = ->l2a. (5.21) Substituting (5.21) into (5.19) gives (4y9'-4yff + l)fl'-2/i' =0. (5.22) The above equation yields two solutions for h, here we only take the positive solution h = -={\-2P)a. • (5.23) 63 Therefore, when b = -Jla and h = -7=(1 - 2fi)a , the redundant 4-8 parallel manipulator is at the maximum quality index as shown in Figure 5.3, and from (5.15) ^JMJl = (VdetJJ^)_ = 4yf2a' (5.24) where Jm denotes the Jacobian matrix of this configuration. It is interesting to note that this maximum value of Vdet JJ ^ is independent to the value of fi. Figure 5.3: Plan view of the optimal configuration of the redundant 4-8 in-parallel manipulator with the maximum quality index From (3.24) and (5.24), it shows that both the redundant 4-4 and the 4-8 parallel manipulators have the same maximum value of VdetJJ^ . Figure 5.4 illustrates the compatibility of these two results. It can be observed that as the distance between the pairs of separation points of the double ball-and-socket joints E, F, G and H of the original 4-4 manipulator increases, the height h at which the manipulator has the 64 maximum quality index decreases (see (5.23)) from h = —i= {P= 0, concentric ball-and- V2 socket joints) toh = 0(/3= —, platform is degenerate). Figure 5.4: Compatibility between the redundant 4-4 and the 4-8 parallel manipulators ^ - V2 ^ 5.2 Implementation From (1.2) and (5.24), the quality index for the redundant 4-8 parallel manipulator shown in Figure 5.1 can be expressed as VdetJJ^ ■• ■ 'I /l=- 4V2fl- (5.25) In this section, a redundant 4-8 parallel manipulator with a = 1 and b = 42 \s, taken as an example for the investigation of the variation of the quality index X during a number of platform movements. 65 The first platform movement we studied is a pure vertical translation of the platform along the z-axis that passes through the center of the platform. From (5.15) and (5.25), the quality index for this movement becomes X = - [a^ -yl2ab + {2/3^ -2j3 + l)b^ +2h^\ With a = 1 and b = V2 , this reduces to 16V2(l-2y9)'/i^ (5.26) X = [i\-2J3f +2h^Y (5.27) and is plotted in Figure 5.5(a) as a function of h and fi. Figure 5.5(b) plots the variation curves of the quality index for several different values of J3. From these figures, we can see the height (hm) at which the manipulator has the maximum quality index is reduced as fi increases. Each value of fl designates the distance between the separation points in the base and is a first design parameter. Clearly, fi=0 is the best overall design. (a) (b) Figure 5.5: Quality index for platform vertical movement 66 The second platform movement is a pure horizontal translation of the platform away from its initial location at height h. To derive an expression for -^detJJ^ , we assume the center of the platform move to point (x, y, h), then the coordinates of the points A, B, C, and D on the platform become X y- Sa h ( 4.^« U X y^ h B D x + 42a 2 r V2a X-- y h y h (5.28) The coordinates of points Ea, Fa, Fb, Gb, Gc, He, Ho, and Ed on the octagonal base can be found from (5.2). Calculating the Plucker line coordinates for each of the eight leg lines yields 5, = , , Sa-b , bh ^ , bx + djSa-ly) (5.29) S,= . Sa-b , bh ^ , bx-dJ-Jla-ly) x-d^, y , h; - — , -d^h, *- ^ (5.30) S3 = „, -Ha-b ^ , . . . bh d,(42a + 2x) + by x + — - — , y + d^, h; -d,h, - — , -^ — - — ^ 2 2 2 (5.31) "^4 = ^Sa-b , , , , bh d.(42a + 2x)-by x + — - — , y-d^, h; d^h, - — , — *- — '- — ^ 2 2 2 (5.32) Ss = , ^y[2a-b J bh ^, bx-dA42a + 2y) x-d^, y + — , h; — , -d^h, i^^ ^ (5.33) ^6 = ^ , ^42a-b , bh ^ , bx + dAyl2a + 2y) x + d^, y + , h; —, d^h, ^ ^ (5.34) 67 S,= Sa-b , , ^ , bh d,{42a-2x)-by X , y-d., h; d.h, — , 2 2 2 (5.35) ■^8 = 42a-b ^ u ^^ X — , y + d^, h; -d^h, —, d^ (V2a - 2jc) + by (5.36) where same as (5.2), d^ i\-2j3)b The Jacobian matrix can then be constructed by using (5.11). Further, ^det JJ^ becomes j^-—^ a'b'h'jX-ipf^ljl^ +l^+l^+l^)(l', +ll +ll +ll) (5.37) where the leg lengths are l,=J{x + dJ + + h\ l,=J{x-dJ + h=- x + - 42a-b + h\ + {y + dj+h\ l,=. x + - Sa-b \2 + {y-dj+h\ ls=J{x-dJ + ln=- X — - ^a-b l,=Mx + dJ + + {y-dj+h\ /,=. X-- yfla-b 2 , 1.2 + {y + dj+h With a = 1, Z7 = V2, and /i = /z„ = ^^ p^^ = ^-^ , from (5.25) and (5.37), the quality index becomes 68 , JdetJJ^ (1 - 2B)\l^ + ll + 1} + ll ) ^= I, , ,, = ,,2,2,2,2 (5.38) where /, =le =4x^ +42{\-2l3)x+y^ +{\-2p)\ 12=15= V^' -S(l-2j3)x+ y^+(l-2fi)\ h=k= V^' + V2(l -2l3)y + y^+{\-2p)\ 1,=It =^x' -yf2{l-2j3)y + y' +a-2J3)\ In Figure 5.6, the quaUty index and its contours as the platform is translated away from the central location while remaining parallel to the base at h^, are drawn for various values of y9. It should be noted that when fi=0, the 4-8 manipulator becomes the 4-4 manipulator and its corresponding quality index is drawn in Figure 3.5. Comparing Figure 5.6(a)-(d) and Figure 3.5, it is clear that the smaller fi, the larger workspace area of the platform is with high quality index. Now we attach a new coordinate system x'y'z' to the square platform. This coordinate system may be obtained from the platform configuration shown in Figure 5.2 by initially aligning it with the xyz coordinate system on the base and then raising it by a distance h along the z-axis to the top platform. Thus, the x'- and >''-axes are parallel to the X- and y-axes respectively when the platform locates at its initial central position shown in Figure 5.2. We are interested in deriving -,/detJJ^ when the platform rotates about the x'- and y'-axes from its central position. Here, we only derive the platform rotation about the 69 y'-axis. But the result to be derived is the same for the platform rotation about the x'-axis since the redundant 4-8 parallel manipulator is fully symmetric. 1.0- 0.5 >.0.0 -0.5 - -1.0 1 -1 (a) ;& = -at ;i„= — -0.14 '^ 5 ""10 -1.0 -05 0.0 0.5 1.0 X 1.0 - 0.5 - ?^0.0 - -0.5 -1.0 - 1 -1 1 J? (b) I5 = - at /i„ = ^^ = 0.24 3 " 6 70 1.0 - 0.5 P^O.O -0.5 -1.0 1 -1 4 4 1.0 - 0.5 PnO.O -0.5 - ■1.0 T r 1 -1 ■1.0 -0.5 0.0 0.5 1.0 (d) fi = - aih= — ^0.53 ^8 "■ 8 Figure 5.6: Quality index for platform horizontal translation with different values of fi i^ ■ " "'? T 5 f: *■• 71 Figure 5.7 illustrates a side view of the moving platform ABCD rotated dy about the _y'-axis. The coordinates of the vertices A, B, C, and D become -Jla ^ H \ / h B J V \ / t D > V ■v2(3 -v2a cos^„ h-\ sin^„ 2 • 2 ' \ cos^^ h sm^, 2 ' 2 ' (5.39) and the coordinates of vertices Ea, Fa, Fb, Gb, Gc, He, Ho, and Ed on the base can be found from (5.2). Fb(Gb) Figure 5.7: Platform rotation about the ^^ '-axis Note that the positions of Une $i, $2, $5, and $6 do not change during this platform rotation and their corresponding Plucker line coordinates can be obtained from (5.3), (5.4), (5.7), and (5.8) respectively. The Plucker coordinates for the line $3, $4, $7, and $8 are now given by 72 53 = ^!la COS e^ - b 42a sin 9^ + 2h 2 2 d^ {42a sin 6^ + 2/i) b{42a sin ^, + 2/z) 42ad^ cos ^, (5.40) ^4 = 42a cos ^^, - ft 42a sin 6'^ + 2ft d^ {42a sin 6'^. + 2h) b{42a sin 6*^ + 2h) 42ad^ cos ^^ (5.41) 5,= V2fl cos ^^. - b , -d^, -■ 42a sin 6^ - 2h 2 ^ 2 d^ {42a sin 6^. - 2h) b{42a sin 0^ - 2h) 42ad^ cos d^. (5.42) 5« = v2a cos ^y - £> 42a sin ^^ - 2/i ~ > ^4> T > d^ {42a sin ^^, - 2h) b{42a sin <9^ - 2h) 42ad^ cos ^ (5.43) with d^ = (l-2y^)fe From Figure 5.7, we have li =1^ =1^ =1^=1 , 1^=1^, and l-j=l^. The corresponding Jacobian matrix can now be obtained by (5.1 1), and further -^det JJ^ can be determined. With a = \,b = V2, and h = h^= — iJ- , from (5.25), the quality index 42 then becomes 73 cos 30^ ) + (16y3' - 32/3' + lAp^ -Sfi- 2)icos20, + 1)) + (Z' + /' )(14y9^ 6/? + l + 8y5'(y5-2)(cos2^y +l) + 2y3(5y9-l)cos2^y) + (/3' -l,)(S/3' - I2j3^+6J3-I)sin2ey2 (5.44) where l = l-2J3, Zj = ^4y9^ - 4;5 + 2 - cos ^,, - {2/5 - 1) sin 6^ , /7=^4/?'-4y5 + 2-cos^^+(2y0-l)sin^^. Figure 5.8(a) plots the quality index as a function of ^ and /3. Figure 5.10(b) presents the change of the quality index for several different values of fi. (a) 90 -60 -30 30 60 Rotation Angle 6 (degree) (b) Figure 5.8: Quality index for platform rotations about the x'- and >''-axes 74 A plan view of the redundant 4-8 parallel manipulator with the moving platform ABCD rotated 6z about the z-axis is shown in Figure 5.9. The x and y coordinates of the vertices A,B,C, and D become x^=r sin 6^ , Xg = r cos 6^ , X(; =-rsin0^, x^ =-rcos0^. y^=-rcose^, yg =rsind^, y^=-rsme^ (5.45) where r = V^ The complete set of coordinates of points A,B,C, and D are therefore M^A Va h), B{xg yg h), C{xc yc h) , D{xj, y,, h) where h is the height of the moving square platform above the octagonal base. (5.46) H Hd ifc y =^ Gb /3 / ^ - - .^ _^ r 1 ^vT ^ i T,.^4 l/\ X -t 1 yiJ^^ rt\W \ ■n 1/ \ ^ \ f A"" - - -. ^ ^/ Fb F E Ea 1? Fa Figure 5.9: Platform rotation about the z-axis 75 The coordinates for the corresponding lines $i, $2, ..., $8 are then given by S^ = x,+d^, yA+—, h; — — , d^h. bix,-y,+2fiy,) (5.47) 5,= b bh ^A-d^, yA+TT' h'^ — —^ -d^h. bix, + y,-2j3y,) (5.48) S,= b _, , . , . bh b{xg-2fixg+yg) ^B--' ya+d^, h; -d^h, - — , ^^ (5.49) ^4 = b A u A u bh bjxg-ipxg-yg) ^B--' yB-d^, h; d,h, -—, (5.50) ^5 = b , bh b{Xc-yc+2J3yc) ^c-^4. yc~2' ' Y' "^ ' 2 (5.51) Se = b . bh . . b{Xc + yc~ '^fiyc ) Xc+d^, yc-T' ^; Y' d^h, - (5.52) 5, b , , . , bh ^D + 2' ^o~^4' h- d,h, —., biXa-iPx^ + y^,) (5.53) S,= V ^^ „ -i-w A- ^h ^^ b{x^-ipx^-yi,) ^D + 2' ^o+«4. h\ -d^h, — , (5.54) From Figure 5.9, we have l^ =1^ =1^ = /, and l2=h -h-h- ^^e corresponding Jacobian matrix can now be determined by (5.1 1), and further ^det JJ^ becomes I 4V2aV/i'(l-2/?)^|cos6> I VdetJJ^= ) ^' ' ^ (5.55) where /, =-^l(^[2asme^ +b-2fibf +^2000^8^ -bf +4h^ , k =-i(42asme^ -b + 2fibf +y2acose^ -bf +4h' 76 1 — 2/? With a = \, b = yfl, and h = h„= — ^ , from (5.25) and (5.55), the quality index V2 becomes A = - (l-2/?)'|cos^^ 3 • (2(2y5' -2yff + l)(cos^^ -2)cos^^ +16/?' -32y9' +28y3' -12y9 + 3> (5.56) This is plotted in Figure 5.10(a) as a function of yff and 6^. It shows how the quality index varies as the platform is rotated about the vertical z-axis through its center. The eight legs are adjusted in length to keep the platform parallel to the base at a distance /im- Figure 5.10(b) illustrates the variation of the quality index for several different values of fi. It is shown in these figures that the manipulator has the highest quality index A, = I when 6z=0°, and X = (singularity) when 0z=± 90°. As can be seen in Figure 5.10(b), a slight change of 0z under a large fi has a much greater impact on the quality index than that of the same change under small fi. 1.0 0.8 X ■D 0.6 c § 0.4 O 0.2 A 0.0 -90 -60 -30 Rotation Angle 6 (degree) (a) (b) Figure 5.10: Quality index for platform rotation about the z-axis 90 77 Again, from Figures 5.8 and 5.10 we can see clearly that better designs are obtained as fi reduces to zero. Hence the best 4-8 parallel manipulator design is obtained when the pair of base joints are as close as possible. Since from (5.55), ^/detjj^=0 when ^^=±90°, a redundant 4-8 parallel manipulator always becomes singular when its platform rotates ^^=±90° about z-axis from its central symmetric position. Figure 5.11 illustrates the singularity position when a=90°. Figure 5.11: Plan view of the singularity position of the redundant 4-8 in-parallel manipulator when dz = 90° In complete analogy with the redundant 4-4 parallel manipulator presented in Chapter 3, when ^^=±90°, the moving platform of the redundant 4-8 parallel manipulator can move instantaneously on a screw reciprocal to the eight leg forces on the z-axis with pitch h = + -Jlab Ah This is because for ^z=±90°, from (5.47) through (5.54), the 78 component of moments about the z-axis for each of the eight legs all are equal to CHAPTER 6 THE FORWARD KINEMATIC ANALYSIS OF THE SPATIAL REDUNDANT 4-8 IN-PARALLEL MANIPULATOR The kinematic analysis of the redundant 4-4 parallel manipulator has been performed in Chapter 4. It is shown that the inverse kinematics is straightforward for parallel manipulators while the forward kinematics is difficult. In this chapter the forward kinematics of the redundant 4-8 parallel manipulator shown in Figure 5.1 is studied. It will be shown how this problem can be easily solved by transferring it to the corresponding redundant 4-4 case which then can be solved by using the method presented in Chapter 4. 6. 1 Forward Kinematic Analysis In Figure 6. 1 , coordinate systems 1 and 2 have been attached to the fixed base and the moving platform of a redundant 4-8 parallel manipulator, respectively. The origins of the coordinate systems 1 and 2 are located at points E and A receptively. The objective of the forward kinematic analysis of the redundant 4-8 parallel manipulator is to find the position and orientation of the moving platform given all the eight leg lengths. Eight dash lines connecting the moving platform and the base are drawn in Figure 6. 1. K we connect the platform with the base by legs along these dash lines to replace the original legs, we obtain a redundant 4-4 parallel manipulator with the platform location same as the original 4-8 manipulator. Thus, if we can determine the distances, /o, (/= 1, 2, ..., 8), between the platform and the base along the dash lines, the forward kinematic problem will have been solved by using the method presented in Chapter 4 for the 79 80 redundant 4-4 parallel manipulator. The objective now is to determine values for Iq\, Iq2, . . . , /o8 from the original leg lengths li,l2,...,k. Figure 6. 1 : Coordinate systems of a redundant 4-8 in-parallel manipulator Observing Figure 6.1, we find l\, h, hi, and /02 are in the same plane defined by points A, Ea, and Fa while Ij, U, /03, and /04 are in the plane defined by points B, Fb, and Gb, I5, k, hs, and /06 are in the plane defined by points C, Gc, and Hq, and /?, k, /07, and /qs are in the plane defined by points D, Ho, and Fd. Thus, /q/ can be determined from /, in the same plane. For example, in the plane defined by points A, Fa, and Fa as shown in Figure 6.2, we have W^' :'-lf\.: ■' . : ;■ ^' 81 /o, = 4P^b^ + 1^ - '^fibl^ cos(;r -?),), /o2 =^(l-fi)'b'+lf-2(\-/3)bl,cos(p,. (6.1) (6.2) where cos 9), = 2^(1 -2y9)/, Figure 6.2: Leg relations Similarly, the other leg lengths can be obtained /o3 = Vi^VT/^^^2^W^"cos(^^^^, /o5 = -^P^b"- + /j' - Ipbl, cos(;r - 9^3 ) , (6.3) (6.4) (6.5) 82 loe = yl (^ - fi)' b'+l's- 2(1 -/3)bl, cos (p,, (6.6) /o7 =V^V+/f^^2y9w7cos(^^^^, (6.7) l^= 4{^- Pfb^ +It -2{\- p)bl, coscp, , (6.8) where (1- -iPfb'+ll-ll 2b{\-2fi)l, _o- -2Pfb'+ll-ll 2b {\- 2 15)1, (1- nr\e m — -2pfb'^l^-ll 2b {\- 2 15)1, Using the values of ki, I02, ..., /os as input leg lengths to the forward kinematic analysis presented in Chapter 4, the position and orientation of the moving platform will be determined. 6.2 Numerical Verification A numerical example is presented for a redundant 4-8 parallel manipulator to verify the analysis. The dimensions of the manipulator are measured in an arbitrary length unit and given as follows: platform side a = 10, base side b= 15, B =—. 8 A set of leg lengths are given as /, =12.21787, /^ =9.15596, /, =12.83105, /^ =7.52035, /, =13.47917, /6= 13.13367, /,= 13.88865, /« =14.04687. 83 Thus, the input leg lengths for the forward analysis of the corresponding redundant 4-4 parallel manipulator are obtained from (6. 1) through (6.8) and /o, = 13.59387, /o2 = 9.87590, l^, = 9.87590, l^^ = 7.94680, /o5 = 14.41631, /o6 = 13.98464, l^^ = 14.72302, l^s = 14.92181. The numerical results are presented in Table 6.1. It has been verified by an inverse kinematic analysis that all solutions reproduced the correct leg lengths. Table 6.1: Numerical results of the redundant 4-8 in-parallel manipulator No. 0\ (deg.) O2 (deg.) 0i (deg.) *Pa 'P 'P 'Po 1 -112.939 122.570 -28.524 "10.409" 3.408 _ 8.052 "14.940" 12.304 . "7-475 _ ' 7.091" 16.592 11.948 " 2.560] 7.696 12.525] 2 112.939 -122.570 28.524 '10.409" 3.408 -8.052 "14.940' 12.304 _-7.475_ 7.091' 16.592 -11.948 2.560] 7.696 Similar to the redundant 4-4 parallel manipulator, there is a special solution when the platform rotates about the z-axis (Figure 5.9). For example, when the leg lengths now become /, =/3 =/5 =/^ =/ = 18 and 1^=1^ =1^=1^ =l'= 16 for the redundant 4-8 parallel manipulator in the first example, two configurations above the base plane with another two reflected through the base are obtained as shown in Table 6.2. 84 Table 6.2: A numerical example for the special case of the redundant 4-8 in-parallel manipulator No. h Pa 'P P P ■3.022" ■-6.393' "-3.022" " 6.393' 1 7.498 6.393 7.498 3.022 -6.393 7.498 -3.022 _ 7.498 " 3.022" ' 6.393' "- 3.022" "-6.393" 2 15.748 -6.393 _15.748_ 3.022 _15.748_ 6.393 15.748 -3.022 _15.748_ " 3.022' "-6.393" "- 3.022" " 6.393' 3 -7.498 6.393 -7.498_ 3.022 -7.498 -6.393 -7.498 -3.022 -7.498 3.022' 1 6.393" 1 ■ -3.022" " -6.393" 4 -15.748 -6.393 3.022 6.393 -3.022 -15.748 ■ -15.748 -15.748 -15.748 CHAPTER 7 THE OPTIMUM QUALITY INDEX FOR A SPATIAL REDUNDANT 8-8 IN-PARALLEL MANIPULATOR A redundant 8-8 in-parallel manipulator is shown in Figure 7.1, which is derived by separating the double ball-and-socket joints in the base and the top platform of a 4-4 manipulator shown in Figure 3.1. A plan view of the redundant 8-8 in-parallel manipulator is shown in Figure 7.2. The device has eight legs connecting an octagonal platform and a similar octagonal base. The octagonal top platform is formed by 4 pairs of joints Ai and A2, fii and B2, C\ and C2, and D\ and D2. Each pair of joints is separated from a vertex of a square of side a by a distance cea for which 0<«r<— . Similarly, the octagonal base is formed by 4 pairs of joints En and £a, F/^ and Fb, Gb and Gq, Hq and Hd, and each of them is separated from a vertex of a square of side fc by a distance fib for which 0<P<—. This design has the distinct advantage that it completely avoids the mechanical interference problem associated with the design of double spherical joints. 7.1 Determination of ydet.T^.T^ ' m The moving platform of the redundant 8-8 parallel manipulator shown in Figure 7.2 is located at its central symmetrical configuration and is parallel to the base with a distance h. It will be shown in this chapter that a maximum value of the square root of the determinant of the product of the manipulator Jacobian by its transpose, i.e., ^detJ^J^ , may be obtained from this symmetric configuration. 85 86 Figure 7. 1 : A redundant 8-8 in-parallel manipulator Figure 7.2: Plan view of the redundant 8-8 in-parallel manipulator First, a fixed coordinate system is placed at the center of the octagonal base as shown in Figure 7.2. Then the coordinates of the points Ai, A2, fii, B2, Ci, C2, D\, and D2 on the platform are where f A fi, -d^ flj h V (4io -d, -Jj h 42a - Jj h D, V2a ^5 — z- ^5 h ) B. D. ' ' 2 V 2 ' ' J 5 2 5 V h J 5 2 5 V h J 42 a a (7.1) 87 The coordinates of the points Ea, Fa, Fb, Gb, Gc, He, Ho, and Ed on the base are ■^ 0^ , F, b \ „(b , F, 2 -d. ^ . G, fb d. f b \ d, - 2 ^ ) \ , HA-d, - . ^D / b \ (7.2) T d, E — -d. Same as (5.2), d^ - {\-2P)b Now the Pliicker Une coordinates of the eight leg lines of the redundant 8-8 parallel manipulator can be obtained by counting the 2x2 determinants of the various arrays of the joins of the pairs of points EaAx, FaAi, ..., EdD2. From points Ea and Ai, we get "' / ■ ^ d^-d^, d.-d,, h; bh , d.h, d. where d^=—(2ap-2a-2p + \)ab and d.^ "" ^ 4 2 Similarly, from points Fa and Ai, = ^5-^4. bh From points Fb and B\, S,= ds-di, h; -^, -d^h, -d^ it bh d\-d^, d^-ds, h; -d^h, — — , d^ From points Gb and Bi, (7.3) (7.4) (7.5) ^4 = bh d\-d^, d^-d^, h; d^h, - — , -d^ (7.6) From points Gc and C\, ■^5 = 88 hh d^-d^, d^-d^, h; —, -d^h, d^^ (7.7) From points He and C2, V 5^ = hh d^-d^, d^-d^, h; —, d^h, -J^ (7.8) From points Ho and Di, S,= bh d^-d^, d^-d^, h; d^h, — , d^ (7.9) From points Eu and D2, 5,= d^-d^, d^-d^, h; -d^h. bh (7.10) The above coordinates are not normalized and each leg line needs to be reduced to unit length. Then, the normalized Jacobian matrix of the eight leg lines can be expressed in the form J = si si Si Si si si Si l^i I2 h K h k h h (7.11) where UJi, ...,k are leg lengths. Here, the device is in a symmetrical position so that the normalization divisor is the same for each leg, namely /, = /2 = . . . = /g = /, and for every leg [77 ~ ; (7.12) = J-[(2a' -2a + l)a' +V2(2a'yff-l)aZ7 + (2y9' -2y5 + l)^' +2/i'J. From (7.3) to (7.10), (7.1 1) becomes 89 -7 d^—d^ d^—d^ d^-d^ d^-d^ d^—d^ d^—d^ d^—d^ d^—d^ d^ - <i, d^— d^ d^ —d^ d^— d^ <i, —d^ d, - d^ d^ —d^ d^- d^ bh 2 bh 2 -d,h d,h bh 2 bh 2 d,h -d,h bh 2 bh 2 -d,h d,h -d. -d. dji bh 2 d. -dji bh 2 -d. (7.13) Using equation (7.13), the determinant of the product JJ^ can be expressed in the form detjr=-l d,+d^ d,h d,+d. -d,h %h^ d^h 2m.2 -d^h {b'+4d;)h Qj'+Ad^W Ml (7.14) where d,^2{2a''-2a + \)a''+^I2{2ap-\)ab, d^ = 2(2y9' -2/3 + \)b^ + V2(2ar^ - l)ab. Expanding (7.14) and using (7.12), then extracting the square root yields / , „T 256y/2\2aj3-2a-2/3 + l\\'b'h' Vdetjj'^ = L_^- ^^— J . (dT+d^+4h^)^ (7.15) It is important to note that VdetJJ^ = (with a,b,h^ 0) when 90 ■ fia,fi) = 2aJ3-2a-2fi + \ = 0. (7.16) The plot of equation (7.16) is illustrated in Figure 7.3. Four cases that satisfy the relation f{a,fi) = are also drawn in Figure 7.3. When a and ;5 satisfy the relation f{a,fi) = 0, a redundant 8-8 parallel manipulator has finite mobility even if all its eight actuators are locked. A similar relation was found for a nonredundant 6-6 parallel manipulator by Lee and Duffy [20] and this kind of mobility was discussed in Hunt and McAree [14]. It is interesting to note that the two end points of the curve f(a,fi) = 0, i.e., a = 0, y9 = - , and a=—,/3= 0, are two cases for which the manipulator is degenerate. Figure 7.3: Plot of f(a, j3) = 2afi-2a-2J3 + \ = 91 Now, assuming the top platform size a is given and taking the partial derivative of (7.15) with respect to h and b respectively and then equating to zero, we obtain 76sS\2a/3-2a-2j3 + \fa^b^h\dT+dg-4h^) and = (7.17) 16S^f2\2aj3-2a-2fi + lfa^b^h'id^-dg+4h^)_ T — A = 0. (7.18) (dT+d,+4h^)' When a, b, and h are not equal to zero and 2afi -2a-2fi + l^0, from equations (7. 17) and (7.18), we get d, +d, -4h^ = 2[(2a' -2Qr+l)a' +42{2ap-\)ab + (2p^ -2p + \)b'' -2h']=Q, (7.19) d, -rfg +4/i' = 2[i2a^-2a + l)a^ -{2/3^ -2j3 + l)b^ +2/i']=0 . (7.20) Adding above two equations gives 2dT=2[2i2a^-2a + l)a^+S(2a/3-l)ab\=0. (7.21) Then solving (7.21) yields ^ V2a(2ar^-2a + l) "' i-2ap ■ <'-^^' Substituting (7.22) into equation (7.21), then solving for h we can get two solutions, here we only take the positive solution (again, it should be noted that the negative solution is simply a reflection through the base) , \2afi -2a-2p + \\aJ2{2a'' -2a + \) h=i—!- "- — ^1^ i. (7.23) 2{\-2al3) Clearly, h=Q when f{a,p) = 2aP - 2a - 2y5 + 1 = . The variation of h with respect to a and y5 for a = 1 is shown in Figure 7.4. 92 0.5 Figure 7.4: Plot of h vs. or and y5 with a = 1 Finally, substituting (7.22) and (7.23) into (7.15) yields VdetJ^Jl =(VdetJJ^)_ ^4y[2(2a' -la + l^a' (7.24) where Jm denotes the Jacobian matrix for the configuration at which VdetJJ^ has a maximum value. It is interesting to note that this value is a function of a only and not of fi. When a=0, V^et J„ J^ has a maximum value 4V2a' , and when a= - , -^detj^j]^ becomes minimal, 2a\ Figure 7.5 plots the variation of ^detj^j^ with respect to a for a= 1 and shows that VdetJ^jT decreases as or increases. The redundant 8-8 platform with a=- and y9=- is shown in the optimal 8 8 configuration in Figure 7.6 and for which, from (7.22) and (7.23), b=—42a and 31 85 I 195 h ^—rza . From (7.24), the corresponding value of Vdet Jj"^ is -—a . 248 32 93 -E 4 1 - k V- .• 4 J..... ^^. •. .' i J..... 0.0 0.1 0.2 0.3 0.4 0.5 a Figure 7.5: Plot of -y/detj^jj, vs. orwith a = 1 Figure 7.6: An example of redundant 8-8 manipulator in optimal configuration 1 . 25 with a=-,p=-, b= — ^f2a , and h 8 ' 8 31 85 248' 94 7.2 Implementation Now, from (1.2) and (7.24), the quality index for the redundant 8-8 parallel manipulator is given by V_' '- ' Vd etlJ"^ '^--— 3—- (7.25) 4V2(2ar^-2a + l)2a^ The variation of the quality index now is investigated for a number of simple motions of the top platform. Here, we simply consider the case with a=y^and the top platform side a = 1. The base side ^ is then determined by (7.22). Firstly, consider a pure translation of the platform from the initial position along the z-axis while remaining parallel to the base. For such movement, from (7.15) and (7.25), the quality index is given by , _ 64\2afi-2a-2J3 + \fb^h^ ^ T- (7.26) I — 2 With a=/3, a= I, and b = from (7.22), dy becomes zero and ds reduces l-2a to . 2(2a^-2a-i-l)(l-4Qr-i-2ar')' ''■= SS^rip ■ P-27) Thus, the quality index becomes 3 .3 16V2|2a^-4a-i-l| (l-2a^y(2a^ -2a + \yh^ [(2a' -2ar + l)(2a' -4flr + l)' +2(l-2a^fh^] and is plotted in Figure 7.7(a) as a function of h and a. Figure 7.7(b) presents the change of quality index for different values of a. The variation of quality index with a(=y3) and 95 h is interesting. When a{=P)=0 we obtain the 4-4 platform shown by Figure 3.2, when a(=/J)=— we obtain another 4-4 platform shown by Figure 7.8, when ai=fi) is the solution of (7.16) for which a=fi= 1 — j=, the platform is degenerate and /i = (see also V2 Figure 7.3). It follows that as a increases from to 1 — ;= the value of h for /l=l V2 decreases whereas when or increases from 1 — pr to — the value of /i for A= 1 increases. V2 2 Each value of or designates the distance between the separation points in the top platform and base and is our initial design parameter. Clearly, cir(=y9)=0 is the best overall design. We now derive an expression for ^det Jj'^ when the platform of the redundant 8-8 parallel manipulator is translated away from its central location parallel to the base with height h. Assume the center of the moving platform to move to point {x, y, h), then the coordinates of the points Ai, A2, Bi, B2, Ci, C2, Du and D2 on the platform become x-d^ y + d^- V2a h B,\x-d,+^ y-d, h x + d^ y + d^- Sa 5. J x-d^-V y-^d^ h J x + d^ y-d^ + 4ia x-d^ y-d^ + Sa .1 (7.29) D, x + d^- y + d^ h D. x + d, — y-d, h 96 0.3 2 p^a 1.0 - ::::;( T X^XV/'^^x ■ aAA \ \p=a 0.8 - A/ A \ \ '^ aA \ IV''^""^ ^ X o T3 C 0.6 - il • \ v\ U«=2^5 /A: \ \ .3v^ ir ||/\; \ 3t^\ d=i/4 m 0.4 - 1/ / V V''1\ vy^ i ..3 o ll/ \ \ ^ ^X: \J/"~ U8 0.2 - Pl^^^ ^a=0 1 1 1 "! ' 3 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Height h (a) (b) Figure 7.7: Quality index for platform vertical movement Figure 7.8: Reduction of the size of the redundant 8-8 in-parallel manipulator when a{=P) = — 97 The coordinates of points Ea, Fa, Fb, Gb, Gc, He, Ho, and Ep on the base can be found from (7.2). Thus, the Plucker line coordinates for each of the eight leg lines are determined as S,= ^ , , . ^ . . ^^ ^ , b(x-d^)-d^(2y-^f2a + 2d,) x+d^-d^, y-di+d^, h; , d^h, ^ 1L_£ il , (7.30) S,= x-d^+d^, y-di+d^, h; -—, -d^h, ^- *-^^ ^ ^ 2 , (7.31) 53 = x+di-d^, y+d^-d^, h\ -d^h, bh d^{2x + ^[2a-ld^)+b{y-d^ JC + . , J ^ J I, J I. bh d,(2x+^/2a-2d,)+b(y+d,) di-d^, y-d^+d^, h; d^h, -—, — ^ ^- — ^-^ ^ . (7.32) , (7.33) ^5 = w ^^ ^^ ^ u ^^ ^ u b{x+d,)-d,{2y+^[2a-2d,) x-d^+d^, y+d^-d^, h; — , -d^h, *-^ ^— , (7.34) ■^6 = x^d,-d„ y^d,-d„ h- ^, dA J^--d.)^d,i2y^^a-2d,) , (7.35) 5,= ^A ^A A .A u A I. bh d^{2x-42a + 2d,)+b{y+d.) x+d^+d^, y-d^+d^, h; d.h, — , — ^ ^- ^^ ^— 2 2 , (7.36) A ^A .A A I A 1 bh d^{2x-42a + 2d.)-b{y-d.) x-di+d^, y+d^-ds, h; -d^h, —, -^ —^ — ^-^ 5Z 2 2 , (7.37) where du d^, and ds can be found from (3.13), (5.2), and (7.1) respectively. The above coordinates are not normalized and each row must be divided by the corresponding leg length. The Jacobian matrix can then be constructed by using (7.11). Further, ^detJJ^ becomes 98 II 1 1 1 1 1 1 where the leg lengths are l,=^{x + d,-dj +{y-d^+d,y +h\ l^=^{x-d,+d,f +{y-d,+dj +h\ l^ =^{x + d,-dj +{y + d,-d,y +h\ l^=^{x + d,-dj+{y-d,+dj+h\ l,=^{x-d,+dj +{y + d,-dj +h\ l^=^{x + d,-d,f +{y + d,-d,f +h\ l^ =^{x-d,+d,f +{y-d,+dj +h\ l^=^{x-d,+dj+{y + d,-d,y+h\ Substituting (7.38) into (7.25) with a=/3, a=l, Z? = ^^^^^' ~^f "^^^ from (7.22), and l-2a 2a^-4a + l h = h_ = ■^2(2a^-2a + l) from (7.23), the quality index for the platform 2(l-2ar0 translation of the optimum redundant 8-8 parallel manipulator will be determined. Figure 7.9(a)-(d) illustrate the quality index surfaces and contours for various values of a as the platform is translated away from the central location while remaining parallel to the base. When a(=fi) = 0, the manipulator reduces to the redundant 4-4 parallel manipulator whose quality index surface and contours have been shown in Figure 3.5. Comparing Figures 7.9(a)-(d) with Figure 3.5, it is clear that cases (b), (c), and (d) are unacceptable designs. 99 1.0 - 0.5 P^O.O -0.5 - •• ■1.0 1 -1 -1^ -0.5 0.0 0.5 1.0 X (a) «(=;?)= 1 at ;i„=l 1.0 -■■■. 0.5 - ?s0.0 -0.5 -1.0 - 1 -1 -1.0 -0.5 0.0 0.5 1.0 (b)a(=;^=|at/.„=^^0.21 100 1.0 0.5 ^0.0 -0.5 -1.0 (c)a{=/J) = - at /i„= — = 0.08 4 ""28 »)\- —I 1 1 1 1— -1.0 -0.5 0.0 0.5 1.0 X 1.0 0.5 ?^0.0 - -0.5 - -1.0 - 1 -1 1 1 1 1 r -1.0 -0.5 0.0 0.5 1.0 1 8S Figure 7.9: Quality index for platform horizontal translation with different values of a (= y9) 101 Figure 7.10 illustrates the side view of the moving platform rotated 6^ about the ^''-axis, which is drawn through the center of the top platform and parallel to the j-axis located in the fixed base. The center of the platform is located at height h. Then the coordinates of the vertices A i, A2, Bi, B2, Ci, C2, £>i, and D2 become f -Jt. -d.cosd^ \-d. h-d. sind,. ( V2a A2 d^cosO^ T- + d^ h + d^sinO,, B, (—-d,)cose^ -d, h + i—-d,)sme^. B. C, I>, ( X V ( 1 V ( \ ( V ( {^-d,)co%e^ d, h + i^-d,)smd^ 4i d^ cos 6^ d^ h + d^ sin d^ -d^cos$y J ,4la 42a \ -( rf,)cos^,, d, h-C 2 ' - ' 2 -J5)sin^^. D. Ma ,Sa -(— — d^)cosey -d^ h-{— rf5)sin^^ (7.39) Applying Grassmann's method, the Plucker coordinates for the corresponding lines $1, $2, ..., $8 are given by S,= d^-d^cosO , -di+d^, h-d^sinO; 2 --b(h-d^smd^), d^(h-d^smOy), J4 {^|2a - 2^5 ) - bd. cos 6^ (7.40) 102 MGb) Figure 7.10: Platform rotations about the j '-axis S,= -d^+d^cosOy, -di+d^, h + d^sinO^; ^u^u. J ■ n ^ J fu, J ■ a ^ d^(yf2a-2ds)-bd^cose^ (7.41) 53 = b - {■yjla - Id. ) cos 9 -, d.-d., h + (V2a -2^/5) sin 0^, ^ ^, (V2a-2rf5)sin^,^ b ^, {42a -Id,) sin 6^ -d,(h+ -), --(h+ '-), bd, - c?4 ( V2a - 2d, ) cos 0^ (7.42) 103 ^4 = b-{42a-2d^)cose^ {■sfla - 2d ^) sin d -, -d^+d^, h-^ y . (V2a-2rf5)sin^ ^ (V2fl - 2^, ) sin ^„ dAh+ '-), --{h+ :^ '-), bd^ - d^ {s2a - 2d^ ) cos 6^. (7.43) •^5 = -d^+d^cos6y, d^-d^, h + d^sindy', d^ ( V2a - 2d. ) - bd. cos ^ -b^h + d^sind ), -d^ih + d^sind ), "'* .^j •'"5^^'^^y (7.44) Se = 1 d^-d^cosd , d^-d^, h-d^s'm6; -bih-d^sind ), d^ih-d^sind ), - d^ (V2a -2d,)- bd, cos 9 (7.45) S,= b-{y[2a-2d,)co%e^ {yl2a-2d,)sm0^ '-, -d,+d„ h ^^ ^ {42a-2d,)sme^^ b , {42a-2d,)smd, 2 . (c IMIIC,, bd^ - d^ ( V2a - 2d. ) cos 6 SJ^^^^y S.= b-{^l2a-2d^)cosdy , d^-d^, h- (\2a-2d.)sm0 (V2a-2rf5)sin^ ^ (^/2a-2d,)sm0^ -d,ih ^ ), -ih ^ -). bd^-dA42a-2d^)co&e^ (7.46) (7.47) 104 Since the configuration of the manipulator keeps symmetric about the ;c-axis during the platform rotation about the j '-axis, from Figure 7. 10, we have /, =l^=^(d^cos9^.-dJ'^+(d^ -d^f+(d^smd^. -hf, I2 ^h = -J(^5 cos^^ - ^4 )^ + (^1 -d^f+ (flfj sin 6^ + hf , '3 ~ M ~ ■ h ~ '■» ~ ■ {y!la-2d^)co^d^-b + {d,-d,f + \[2a-2d V -sin^„ +h (7.48) V {42a-2d,)cosd^-b^ ^^ ^. (42a-2d, . + (^4-^5) + ^-sind-h 2 ' J The corresponding Jacobian matrix J then can be obtained by (7.11). Substituting -y/det JJ^ into (7.25), the quality index for such platform rotation will be determined and it is too large to be expressed here. Figure 7.11(a) plots the quality index surface with respect to 6^ and fi for the redundant 8-8 in-parallel manipulator with a=p, a=l, h = — ^ — j— ^ -, and h = h = 2a -4a -1-1 ^2{2a^-2a + l) 2(1 -2a^) \-2a' . The change of the quality index for several different values of y0 is shown in Figure 7. 1 1(b). It should be noted that since the redundant 8-8 in-parallel manipulator is fully symmetric, the same result can be obtained for the platform rotations about the ;c'-axis, which is drawn through the center of the top platform and parallel to the jc-axis located in the fixed base. 105 -90 -60 -30 30 60 90 Rotation Angle 9 (degree) (b) Figure 7.1 1: Quality index for platform rotations about the x'- and y '-axes Figure 7.12 illustrates a plan view of the moving platform rotated Ot, about the vertical z-axis through its center, the legs being adjusted in length to keep the platform parallel to the base at a distance h. The x and y coordinates of the vertices A], Ai, B\, B2, Ci, C2, Di, and D2 are given by x^^ =rsin(d^ +</)), Xb, =rcos(0^-(p), XB^=rcos{d^+(p), Xq =-rsin{e^-<f>), Xc, =-rsm(0^+<p), Xi^ =-rcosid^-<p), Xd, =-rcos(e^+<p). >'a, =-rcos(e^-(p), Va, =-rcos(0^+<p), yg^ =rsm(0^-(p), ya, =rsm{e^ +(/)), ^c, =rcos(6>^-0), y^ =-rsin(^^-^), Jd, =-rsm(0^+<p). (7.49) where r-—yl2(2a^ -2a + l) and ^ = arcsin a yl{2a^-2a + \) 106 Figure 7.12: Platform rotations about the z-axis The complete set of coordinates of points Ai, Az, B\, B2, C\, C2, D\, and D2 are therefore Ak, A h), B^[xg^ y,^ h), C^x^^ y. h), D,(x^ y h\ ^k, yA, h\ bJ^x >'fi. h\ QL j. h), dXxd, Jo, h). (7.50) The coordinates for the corresponding lines $1, $2, ..., $8 are then given by S,= b , bh _, , bx. (7.51) ^2 = b , bh , , bx. XA,-d„ >'^,+-, h; -—, -d,h, -^ + d,y^^ (7.52) h hh by ^B,--. yB,+d„ h; -dji, -—, d,x,^+^ (7.53) 107 ^4 = (7.54) ■^5 = bh bXr Xc,-d„ >'c,-^. h; y, -d,h, ^ + ^4>'c, (7.55) Se = Xc,+d„ yc,--' h; —, d,h. bXr -d,yc. (7.56) •^7 = (7.57) S,= ^D^ + j' ^02+^4. ^; -^4^. 2 C/^Xn - ^^D, ' '^4-^D- (7.58) From Figure 7.12, we have 1^=1-^=1^=1^ and l2-h-h-K- The corresponding Jacobian matrix can now be determined by (7.11), and further ^detJJ^ becomes VdetJJ^ =■ A^!2a'b^h\2pa-2a-2p + Xf cosd^ Pi' (7.59) where /, = j(r sin(^^ -(/>) + dJ+(r cos(^^ - <^) - ^"l + /^\ ■ l,=J{rsm(0^+<P)-dJ + ( b\ rcos{6^ +(p) — + h' Substituting (7.59) into (7.25) with a=fi, a=l, b = ^^f^^ — ^^, and l-2a' h = h = 2a^ -4a + l ■^2{2a^-2a + l) 2(1 -2a') , the quality index will be determined as a 108 function of 6^ and a. This is plotted in Figure 7.13(a). The variation curves of the quality index for several different values of a are shown in Figure 7.13(b). -90 -60 -30 30 60 90 Rotation Angle 9 (degree) (a) (b) Figure 7.13: Quality index for platform rotation about the z-axis From Figure 7.13(a) and (b), we can see the manipulator has the highest quality index ;i = 1 when ^ = 0°, and A = (singularity) when 61, = ± 90°. Again, from Figures 7.11 and 7.13 we can see clearly that better designs are obtained as a {=P) reduces to zero. Hence the best 8-8 parallel manipulator design is obtained when the pair of separated joints in the base and top platform are as close as possible. Since from (7.59), -y/detjj"^ =0 when (9z = ±90°, a redundant 8-8 parallel manipulator always become singular when its platform rotates 6*2= ±90° about z-axis from its central symmetric position. Figure 7.14 illustrates the singularity position for ^, = 90°. 109 Figure 7.14: Plan view of the singularity position of redundant 8-8 in-parallel manipulator when dz = 90° Similar to the redundant 4-4 and 4-8 cases, when ^=±90°, the moving platform of the redundant 8-8 parallel manipulator can move instantaneously on a screw reciprocal to the eight leg forces on the z-axis with pitch h^ =H ^^. This is because when ^z=±90°, from (7.51) through (7.58), the component of moments about the z-axis V2 for each of the eight legs all are equal to ± — ab{\ - lap) . 4 CHAPTER 8 THE FORWARD KINEMATIC ANALYSIS OF THE SPATIAL REDUNDANT 8-8 IN-PARALLEL MANIPULATOR In this chapter, the forward kinematic analysis for the redundant 8-8 parallel manipulator shown in Figure 7.1 is studied. Since there are no double-spherical joints in the 8-8 manipulator, the method used for solving the forward kinematics of redundant 4-4 and 4-8 parallel manipulators does not work for the 8-8 case. We use a different method to determine the location of the moving platform given the actuator displacements of all the eight legs. . 8. 1 Coordinate Systems First, two coordinate systems, 1 and 2, have been attached to the fixed base and the moving platform respectively as shown in Figure 8.1. The origins of the coordinate systems 1 and 2 are located at points E and A receptively. The coordinates of points Ea, Fa, Fb, Gb, Gc, He, Ho, and Ed on the base are known in terms of the coordinate system 1 and >" '{\-P)b b >- 'p - *^Fa 'P - il-fi)b 'P - b Ed "0" ' b' ^ 'p = *^Fb - Pb Gc '{\-P)b b 'P - {\-p)b (8.1) 110 Ill while the coordinates of points A\, Aj, B\, Bi, C\, C2, D\, and D2 on the platform are known in terms of the coordinate system 2 and P. = (1- a)a a aa aa ^P - ^P - {\-a)a ^P - a ? ' 't aa a '{\-a)a ' % = aa > 2p _ a ^P - (l-a)a (8.2) Figure 8. 1 : Coordinate systems of a redundant 8-8 in-parallel manipulator 112 The objective of the forward analysis of the redundant 8-8 parallel manipulator is to determine the coordinates of points Ai, A2, 5i, B2, Ci, C2, Di, and D2 measured in the coordinate system 1 given eight leg lengths /, (/= 1, 2, .... 8). Thus, we need to relate the position and orientation of the coordinate system 2 in three-dimensional space to the coordinate system 1. Once this has been accomplished, it is possible to transform the coordinates of any point in coordinate system 2 to coordinate system 1 . Here, we let the nine coordinates of three non-coUinear points from the moving platform be chosen to represent the position and orientation of the coordinate system 2 in space. Then, the coordinates of the other points on the platform are expressed in terms of these nine coordinates. Assume the coordinates of points A, B\, and D2 measured in coordinate system 1 are Pa = "^a' "^b" '^d' ^A 'P = ^B 'P = )'d .^A. .^B. .^D. (8.3) Since vector V^- P^ is along the j:-axis of coordinate system 2 while vector 'Pp -'P^ is along the j-axis, the z-axis will be along ('Pb_ -'Pa) xC'Pd^-'P^) . Therefore, the three unit vectors along the coordinate axes of the coordinate system 2 measured in the coordinate system 1 are 'x =— ^ {\-a)a (8.4) 'P -'P J2 ~ (1 - a)a (8.5) 113 ('Pb-'PJx('Pd-'PJ They specify the orientation of the coordinate system 2 relative to 1 and can be represented as a 3x3 matrix lR = ['x, 'y, 'zj. (8.7) Since point A is the origin of coordinate system 2, the coordinates of points Ai, A2, ^2, Ci, C2, and Di may now be expressed in the coordinate system 1 as 'P,,= 'P.+ lR^, Pflj = Pa + 2^ Pbj » Pc, = Pa + 2^ Pc, ' Pc2 = Pa + 2^ Pc2 ' Pd, = Pa + 2^ Pd, ' where ^P^^, ^P^^, ^P^, ^P^^, ^P^, and ^P^^ are the coordinates of points A 1,^2, ^2, Ci, C2, and D\ measured in the coordinate system 2 and given by (8.2). The forward problem reduces to determine the nine coordinates of points A, Bi, and D2 in the coordinate system 1 such that the moving platform satisfies the distance constraints with the base. 8.2 Constraint Equations Since the eight leg lengths, /, (i=l, 2, ..., 8), have been given for the forward analysis, the location of the moving platform must comply with these constraints. Thus, 114 the geometry constraint equations between the moving platform and the fixed base can be written in the coordinate system 1 as ('p,,-'p,J•('p,,-'p,J=/^ : (8.9) Cp^^-%)-CK,-%J = il (8.10) ']-, ('Pb.-'Pf,)-('Pb,-'I\) = /3, (8.11) ,, . (%^-'PaJ-CPB,-'Pc,) = ll (8.12) ('Pc,-'Po,)-('Pc,-'Pg,) = /5'. (8.13) ('Pc^-'P„^)-('P,^-'Ph^) = /^ (8.14) ('Pd,-'PhJ-('Pd,-'Ph,) = /7, (8.15) ('Pd,-'Pe.)-('Pd,-'PeJ = ^8- (8.16) There are another three constraint equations representing the distance constraints on points A, B\, and Di on the platform, which have been used to define the position and orientation of the coordinate system 2 relative to 1. These equations are written in the coordinate system 1 as ("PB,-'PJ('PB.-'PJ = (l-a)V, (8.17) (•Po^-'PJ-('P,^-'PJ = (l-«)V, (8.18) ('Pd,-'Pb,)('Pd,-'Pb,) = 2(1-«)V. (8.19) Thus, we have 1 1 equations in 9 unknowns {x^, y^, Zk, x^, y^, zq, x^, yn, and zd)- 8.3 Equation Solution Substituting (8.1), (8.3), (8.8) into equations (8.11), (8.10), and (8.16), then expanding and regrouping yield 115 zl=-xl-yl+2(x^x^ + y^y^ + Zj,Zo)-2J^y^+{\-2a + aV+fi^b'-l^, (8.20) Zb = -4 -yl+ 2bx^ + Ipby^ -(I + 13' )b' + 1] , (8.21) zl = -xl -yl+ 2fiby^ - fi'b' + /^ (8.22) Further, substituting (8.1), (8.3), (8.8) and the above expressions for zl, zl, and zl into equations (8.9), (8. 14), and (8. 19) yields ZAZB=-XAX^-y/,ys + l3b(x^-x^) + b(2-a-fi + a/3)x^ + (8.23) biy^-yo+J3ya)-bil-a-2/3 + a/i)y^+q^, ZbZd = -^B^D - >'b>'d + bx^ + fib(yB + yo) + ^2 ' (8.24) o, /h[axn+(l-3a)yr,] ZaZo = -^A^D - >'a>'d + fibx^ + ^ ; -^ + ^3 . (8.25) l-2a where qi, qj, and q^ are known constants and are defined in Appendix B. Now substituting the expression for zaZd in (8.25) into (8.20), the expression for z\ becomes . - >. , - . - « zl = -xl -yl+ 2pbx, + '^'^^^y^' ^"^ + 2^3 + (1 - 2a + aV + P'b' - ll . (8.26) Five equations that are linear in xa, yA, xb, ys, xd, and >'d are generated by substituting (8.1), (8.3), (8.8), (8.21) - (8.26) into equations (8.10), (8.12), (8.13), (8.15), and (8.17): ^l^A +^2)'a +^-3^b +^4)'b +^5^D +^6>'d + ^7 =0 = 12,3,4,5). (8.27) Solving the above five linear equations with xa, yA, ye, xd, and >'d as unknowns, we obtain 116 Xq — —Xq + q^ 1 >'a=— ; -^8+96 ^"^ (8.28) a l-a where the constants are defined in Appendix B. Now, there is only one unknown xq need to be determined. To find the solution for ;cb, we need to construct more equations. Observing the left hand sides of (8.21) - (8.26), we may formulate the following six identities Eq2 =(zI)(Zj^Zd)-(z^Zs)(ZsZo) = 0, Eq^ = (ZdXZa^b) - (Za^dXZbZd) = 0. (8.29) Eq,=izl)izl)-iz^z^)'^0, Eq,={zl){zl)-iz,z^)'=0, Eq,=(zl)izl)-iz^Z^)'=0. Substituting (8.21) - (8.26), and (8.28) into the above identities, we finally get six equations in only one unknown, ;cb: Eq,{x^) = f^Nyxi=0 (/ = 1,2,...,6) (8.30) where the constants are defined in Appendix B. Choosing any 5 from the above 6 equations, we can determine the value of jcb. Here, we select the first 5 equations and they can be represented in matrix form as 117 Ny=0 (8.31) where N = ^14 A^,3 A^,2 A^n A^,o N N N N N JY24 iV23 ^^22 ^'21 ^''20 ^34 A^33 A^32 A^3. ^^30 A^44 ^43 A^42 A^4I A^40 ^54 A^53 A^52 ^5. ^^50 y = 1 The matrix equation (8.31) may be thought of as a homogeneous linear system in five unknowns. The necessary and sufficient condition under which these linear equations have non-trivial solutions is that the determinant of the coefficient matrix N must vanish. The five equations represented by (8.31) may now be rearranged into the form Wx = r (8.32) where W N,, A^,3 yv,2 yv„ "24 ^23 ^22 '*21 N^ A^33 ^32 A^31 A^44 A^43 A^42 A^4, ^54 ^53 A^52 A^5. x = r = -A^, 10 A^, 20 -N 30 -A^ 40 -N 50 Equation (8.32) represents five linear equations in four unknowns. The vector x may be solved for by selecting any four of those equations. The term xq is the fourth component of the vector x and unique value for this term is thereby determined. Corresponding values of xa, ^a, ^b, xd, and ^d can be obtained from (8.28). Finally, z\, Zb, and zd can be determined from (8.22). It should be noted that there are two opposite sign solutions for 118 Za, Zb, and zd, respectively. However, they cannot be combined arbitrarily since they must also comply with the three equations in (8.25). Therefore, only two groups of the solution of za, Zb, and zd are possible and the manipulator has two configurations for a set of given leg lengths. 8.4 Numerical Verification In this section, a numerical example is presented for a redundant 8-8 parallel manipulator to verify the analysis. The dimensions of the manipulator are measured in an arbitrary length unit and given as follows: platform side a = 10, base side ft = 15, A set of leg lengths are given as /, =13.29955, /^ =14.24887, /, =9.77545, /^ =11.25375, /, = 11.60143, /g = 15.41449, /7= 15.63800, /8 = 18.01 133. The numerical results are presented in Table 8.1. A unique configuration may be easily determined by checking the sign of z coordinate of one of the platform joints. It has been verified by an inverse kinematic analysis that all solutions reproduced the correct leg lengths. Same as the redundant 4-4 and 4-8 parallel manipulators, the above solution failed when the platform rotates about the z-axis (Figure 7.12). At such special configuration, the odd leg lengths are all equal and /, = /, = /j = /^ = / . Further, the even leg lengths are all equal and 1^=1^=1^=1^=1'. Therefore, equation (8.9) through (8.16) become linearly dependent. 119 Table 8.1: Numerical results of the redundant 8-8 in-parallel manipulator No. 'P 'P •Pb, '^ 'Pc. 'P 'P 'P 1 " 4.667" 3.644 12.482 r 5.717] 2.987 [11.799] ["12.857] 5.191 [ 8.943 J [13.592" 6.398 |_ 8.912_ [12.333] 13.856 [1 1.518] [11.283" 14.513 [l2.201_ [4.143" 12.309 [15.057 [ 3.408] 11.102 |_15.088j 2 4.667' 3.644 -12.482 r 5.717] 2.987 [-II.799J [12.857] 5.191 [-8.943J [13.592" 6.398 [-8.912_ [ 12.333] 13.856 [-11.518J [11.283" 14.513 [-12.201 [ 4.143" 12.309 |_-15.057_ [ 3.408] 11.102 |_-15.088j Table 8.2: A numerical example for the special case of the redundant 8-8 in-parallel manipulator No. 'Pa, 'P 'Pb, 'Pb. 'P 'P P P "4.135" [2.961" [-4.889" [-5.677" [-4.135' [-2.961' [ 4.889" [ 5.677] 1 4.889 5.677 4.135 2.961 -4.889 -5.677 -4.135 -2.961 8.675 [8.675_ [ 8.675 [ 8.675 _ [ 8.675_ [ 8.675 [ 8.675 _ [ 8.675] " 2.961" [4.135" [ 5.677" [ 4.889" [-2.961" [-4.135" [-5.677" [-4.889] 2 -5.677 -4.889 2.961 4.135 5.677 4.889 -2.961 -4.135 15.715 [l5.7i5_ [15.715J [15.715J [l5.715_ [l5.715_ [l5.715_ [15.715J "4.135] [ 2.961" [-4.889] [-5.677] [-4.135] [-2.961] [ 4.889] [ 5.677] 3 4.889 5.677 4.135 2.961 -4.889 -5.677 -4.135 -2.961 -8.675] [-8.675J [-8.675] [-8.675J [-8.675J [-8.675J [-8.675J [-8.675] 2.961] [ 4.135] [ 5.677] [ 4.889] [ -2.961] [ -4.135] [ -5.677] [ -4.889] 4 -5.677 -4.889 2.961 4.135 5.677 4.889 -2.961 -4.135 -15.715] [-I5.715J [-I5.715J -15.715] -15.715] [-I5.715J [-I5.715J [-I5.715J 120 For example, when the leg lengths become /, =/3 =/j =/^ =/ = 18 and l^=l^=l^=l^=r=l6 for the same redundant 8-8 parallel manipulator as the first example, two configurations above the base plane with another two reflected through the base are obtained as shown in Table 8.2. CHAPTER 9 CONCLUSIONS Redundant in-parallel manipulators present many interests in various applications: increase dexterity, reduce or even eliminate singularities, increase reliability, simplify the forward kinematics, and improve load distribution in actuators. This work has studied several spatial redundant parallel manipulators. The quality index for redundant parallel manipulators has been defined as a dimensionless ratio which takes a maximum value of 1 at a central symmetrical configuration that is shown to correspond to the maximum value of the square root of the determinant of the product of the manipulator Jacobian by its transpose. A quality index has two clear meanings so far. When A=0, a platform is in singular condition and when A = 1, it is in its optimal geometry static configuration. However, when A is neither zero nor one, it is hard to say exactly how much one configuration is better than another. One can not say that a configuration with A=0.8 is twice as good as a configuration with Ji=OA without further analyses. However, a quality index helps in the design platforms by setting dimensions that give best quality index value. Also, it gives an idea of certain designs that must be prevented as they would lead to zero or lower quality indexes. The quality index reflects singularities, and therefore gives an indication of the safe regions within which the manipulator can be maneuvered and controlled. Using quality index, variable motions are investigated for which a moving platform rotates about a central axis or moves parallel to the base. It shows that the wider 121 122 the range of high quaUty index, the better the design of a parallel manipulator. Thus, the quality index can be used as a constructive measure not only of an acceptable operating workspace but also of acceptable and optimum design proportions. Additionally, the redundant 4-4 parallel manipulator contains double-spherical joints. There are eight of them and they are the source of critical practical difficulties since they can produce serious mechanical interference. There appears to be no reasonable alternative than to accept a reduction in the maximum quality index through separation by fairly short distances of some or all of the double-spherical joints. It appears at this time that it may be best to separate the double-spherical joints as per the arrangement of the 8-8 platform shown in Figure 7. 1 in order to avoid the mechanical interference problem while keeping the reduction of quality index in an acceptable range. The forward kinematic analysis of redundant parallel manipulators has been performed for which it is required to determine the position and orientation of the platform given the leg lengths. The scope of future work includes the development of strategies for redundant parallel manipulators for various applications, a comparison of the performance of nonredundant and redundant parallel manipulators and an assessment of the advantages and costs of redundancy. APPENDIX A CONSTANTS FOR THE FORWARD KINEMATIC ANALYSIS OF THE REDUNDANT 4-4 IN-PARALLEL MANIPULATOR A:, = Ibl^s. k^ =2abs^s^ k,=b'-2abc.c.+ll-ll+ll-l^ ^4 ~ ^^'■2^^ k,=a^- labc^ c^ + 1] - ll + ll - 1] k^ =2abl^s^s^ k^ =2a^bc^s^sl -2i „2 ks=ab c^s^^ -2a bc^c^^s^^ + ^kc^s^, '^^^0,^ +"^6^^,^ '^^^^^ + 2a^bc^/^s^^-2abl^sls^^ k,,=-2ab\s^ +a'bc^^s^^ +bl^c^^s^ -bl^c^^s^ + bl'.c^^s^ -bl^c^^s^ + ^12 =«X^^ -2«^\*^ +b%s^ +2abl,c^c^s^ -bl^c^s^ -bl^c^s^ + 123 124 M,3 = 2aWs^sl(2abc^c^ -labclc^ -a^s^ -b^sl +6abc^c^sl -Ibc^y^ - 2ac^l,sl + 2l',sl + llsl + llsl - 2/^J + l^s^ + l^s^ ) M,2 = absl(2a'bc^c^ -2a'bclc^ +4a^bc^l^ -Aa^bcll^-2abc^c^ll+2abclc^ll - 2abc^c^ll + 2abclc^ll + 2abc^c^ll - 2abclc^ll - 2abc^c^ll + 2abclc^ll - 2\a'b^sl ^Aa'bc^c^sl +6ab'c^c^sl +4a^bc^l^sl^ +Sab\l^sl -Sa^ljsl - Sb^l^sl - 6abc^ c^ llsl + 2bc^ l^sl + 2ac^ I'.sl + 1^^ + a^l^sl - 6abc^ c^ lls^ + 2ac^lf,sl -lUlsl +bHlsl -Aabc^c^llsl +2bc^l,l^sl +2ac^lf,sl -lUls^ - lUlsl -aHlsl -b'llsl ^lOabc^c^llsl -Abc^l.llsl -4ac^l,iy^ ^AlUlsl + 72/2„2 . /2/2 2 _/4 2 + ^2,2 2 _ g -,/,^ ^ /2 2 , 2bc I P <!^ + 2aC I l^ <i^ -l^l'^<!^ - lll'sl +lll',sl ^b'llsl -Aabc^c^llsl +2bc^l,llsl -l^sl -l^sl ^lUlsl - iXsl +24a'bX -nab\l,sl -4a'bc^l,sl +4a'bcll,sl +2AaWslsl - Sa'bc^l,slsl-24a'bXsl) Mu = -s^s^(4a'b'+4a'b'-2a'b\c^ -"^^'^'cl^^ -4«'^'c^< -a'bc^l, + 4a^b'^c^l^ - ab*c^l^ + 4a^b'^cll^ - 5a^bH\ - 2a'bc^c^l\ - 2ab^c^c^l\ + Sa^bc^ll + 2>ab^c^ll - 2abc^c^l^ - 4a^b^l^ + 2a^bc^c^ll + 2a^bc^l^ll + ab^c^lj^ -4aWl^ +2ab\^c^l^ +3a^bc^lJ^ +ab^c^l^l^ -a^l^l^ - 2abc^c^llll + ac^lHl - 2abc^c^l^J^ + ac^lf^l] +SaWll - 2a'bc^c^l^ - 2ab\c^ll -4a'bc^l,ll -4ab\lj', +a'l',l', +b'l',ll + 4abc^c^l',l', -bc^lHl - ac^llll^2abc^c^llll -ac^l.lH^ +2abc^c^l^l', -bc^lj'.l', -ac^l.l^l', +l',l^l', - 2abc^c^lt + bc^l^ll + ac^l^ll - l^l^ - 4aWl^ + 2a'bc^c^l^ + a^bc^lj^ + 3ab\l,l^ -bHll^ -2abc^c^lll', +bc^l'j^ -2abc^c^l^l^ +bc^ylllj + ac^yie, -lllll^ + 2abc^c^lie, -bc^lfj^ -ac^ljH^ +l',l'j^ -4aWl^ + 2ab\c^ll +a'bc^l,ll+2ab'c^l,ll-2abc^c^llll+2abc^c^llll -bc^l.l'ji - 125 \2aVc^l,sl -Aa'b^cll^sl +4a^b^iy^ +6a^b^llsl +6a'b^l^,sl - 4ab\l,l^sl -Ma'bHlsl +Sab\l,iy^ +6aWl^sl -4ab\l,l^sl + eaWllsl -Aab'cJJlsl -6a'b'sl -ba'b'sl +Sa'b'c^c^sl +4a'b'clc^sl - .3. 2„ , 2 .2i.2;2_2 '01 ^ ^ "^ ^ ^ 4a'b\l,s'^ -Sa'b\l,s'^ +4a'b%s'^ +6aWllsl -4a'bc^l,llsl + ea'b'llsl -4a'bc^l,llsl -\2aWiy^ +Sa'bc^l,iy^ +6aWiy^ - 4a'bc^l,l^sl +6aWllsl +6a'b'slsl +6a'b*slsl -4a'b\c^slsl + \2a'b\l,slsl +Sa'b\l,slsl -4a'bY,slsl -6a'bY,slsl -6a'b'l^slsl + M,,=5a'b'sl -2a'b\c^sl -2a%\clsl +a'bc^l,sl -4a'b\l,sl -a'bl'.sl + ab'l^^ + 2a'b\c^llsl - 2a'b\cll',sl + 2ab\llsl + a'bc^llsl - abl^sl - a'biy^ +2a'b\c^i:s^ -abl^l^s'^ -abH^s'^ +2a'b\c;^l^s;^ -2ab\l,l^s'^ - a'bc^l,llsl + ablUlsl + abllllsl + a'bllsl + ab'llsl - 2a'b\c^l',sl - 2a'b\cXllsl +4ab\l,iy^ -bc^l'.iy^ -abl'jy^ -abl^iy^ +bc^l,l^iy^ + abiy^ -bc^l.iy^ -a'bl^s;^ +2a'b\c^l,'s;^ -2ab\l,^sl^ +a'bc^l,^s'^ - ab^l^sl +bc^lll^sl +ablie,sl -bc^lj^l^sl -abl'j^sl +bc^l,l',l^sl -abY.sl + 2a^b^c^clllsl -2ab'c^l,l^sl -a'bc^lf^sl + abllllsl + abllllsl -abllllsl + bc^lilllhl ^abllllsl -bc^ylllsl -4a'b'sl +4a'b\l,sl +5a'b'sl - 2a'b\c^sl -2a^b\c^sl -4a'b^c^iy^ +ab\iy^ +a'biy^ -ab'l^sl + ;3_2 ;4„2 „3,,2„2 2l2. /2„2 ;2„2 ab\l^s;^+2a'bc^l^s;^ -abl^^ -a'biy^ +2a'b\c^iy^ -ab\l,iy^ - I'i^i l2/2„2 -3i2„2 2l2, ;2„2 2a'bc^l,iy^ +abl^iy^ -abny^ +2a'b\c^l,\ +ab\lj^s'^ - ;2_2 /2;2„2 2/2„2 2a'bcJXsi -abl'ltsi +acjn:sl +abl I'si -acJJtltst +a'bi:sl + /2i2 2 , „3,,2„2 ''h'^'i'^'h i2»5 '2"5''«>2 4-5''«k, "^«>|'2-4'-5''«>; '■(>''<h .2/2, ;2„2 abTy^ -4a^b\c^iy^ +4a'bc^l,llsl -ac/.llsl -abl^sl +ac^l,llllsl - abllllsl +ac^l,llllsl +abiy^ -ac^l.iy^ -a'bl^sl +2a'b\c^llsl - ab'c^ylsl -2a'bc^ljtlsl + abllllsl + abllllsl -ac^l^lsl -abllllsl + 126 ac^y!lisl - abHlsl + 2a'b\c^llsl - ablllls^ + abl'j^sl - ablUlsl + . abl^llsl -Ua'b'slsl+la'b'c^c^slsl +4a'b\l,slsl +4a'b\l,slsl + a'bllslsl +abHlslsl -la^b^c^lls^sl -2ab\llslsl -la^bc^lls^sl + abl'^slsl +a'bllslsl -2a'b\c^llslsl+2a'bc^lf,slsl +abllllslsl + abHlslsl+2ab\l,llslsl+2a'bc^l,llslsl -abl^l^sy^ -ablll^^^sl - a'bllslsl -ab'llslsl +2a'b\c^iy^sl -4ab\ljlslsl -Aa'bc^l.llslsl + ablUlslsl+ablUlslsl -abiy^s^ +a'bl^slsl -2a'b\c^l^slsl + 2ab\ljl^slsl+2a^bc^lf,slsl -abl^l^slsl -abl^l^sy^ +ablie,slsl + ab'lislsl+2ab\l,l^slsl +abl^l^slsl -ablXs^sl +abl'Xslsl - abl^iy^sl +Sa'b's'^sl -4a'b\l,slsl -4a'b's'^ +4a'b\l,s'^ +Sa'b'slsl - 4a'b\l,sy^-4a'b'sy^ M,,=^a'b'sy^ M23 = -4ab'sl si (a' - 4abc^c^ + 2ac^l^ + 1] - ll + 1] - ll ) M22 = -b^slsl(a^ + 20a^b^ -Ua^bc^c^ -\6a^bc^l^+4a^c^l^ -4ab^c^l^-2aHl - 4abc^c^ll + 4ac^ll + /j - 20^/5 +I2abc^c^l^ - Sac^lJ^ - 21^1^ + 1* + 2a^ll - Uabc^c^ll + ^ac^lf, + 2llll - 2l^l', + ^ - 2a'l^ + I2abc^c^l^ - 4ac^l,l^ - 2llll + 2llll - 2llll + /; - 24aVsl - 24aWsl + 24a'bc^l,sl - 4aY,sl + 24a'b'slsl) M^i = 2bs^s^{-4a'b'+a'bc^c^ +4a^b'c^c^ +2a'bc^l^-2ab\l^ -SaWc^l^ + labc^ll +b\ll -bc^c^l'^+4abHl-2a'bc^c^ll -4abc^l,l^ +a\l,l^ - b'^c^hll+c^lHl +bc^c^lt -c^lj', -4abHl+2a^bc^c^ll +4abc^l,l', - a\l,l', +b\l,l', -c^llll-2bc^c^llll+2c^l,llll +bc^c^lt-c^l,lt + , 4abHl-2a'bc^c^ll-2abc^l,ll -b\y^ +2bc^c^l^l^ -c^kllll - 2bc^c^llll+c^l,llll +bc^c^i:+6aVsl -Sa'b\c^sl +4a'b\l,sl + 127 -2i2„2 -272„2 2/2„2 ,3r2„2 .21.3, 2abXs: -eabntst +6abXs: -6abXst +6a'b'si -Sa'b'c^c^si - -2-'^ '5"^ '6''0, S"^ '<h ^ ^ ^ 4a'bc^l,sl +4ab'c^l^sl +I2a^b^c^l,sl -Aab'^llsl -Aa^bc^c^lls^ - eabHlsl +^abc^lj^sl -lallllsl +6ab'l',sl -Sabc^l.iy^ +2allllsl - 6abH^sl +Aabc^l^llsl -Ga^b'sls^ +Sa^b\c^slsl -UaWc^l^^sl + eabHls^sl -eab'llslsl +6abY,slsl) M,,=a'b'sl +4a'b'sl -4a'b\c^sl +2a'bY,sl -4ab\c^l',sl +bY,sl - 2a'bHlsl +4ab\c^l^sl -2bY,l',sl +bY,sl +2a'bY,sl -4ab\c^iy^ + 2b'lliy^ -2bHlllsl +bY,sl -2a'b'llsl +4ab\c^lisl -2bY,l',sl + IbHUlsl -2b'llllsl +bXsl -4a'b's\ +a'b'sl +4a'b'sl -4a'b\c^sl - 6a'b\l,sl +4a'b\l,sl -4ab\l,sl +2aWl',sl +br,sl +Sab\c^llsl - 2b\llsl -4ab\llsl +bY,sl -2a'b'l^sl +4ab\c^l^sl +2a'bc^l,l',sl - 2b\l,l^sl -%ab\l,llsl +4b'llllsl +4abc^c^llllsl -2bc^l',l^sl +bY,sl - 2hc^hltsl +lllUl +2aVllsl -4ab\c^llsl -2a'bc^lf,sl +2b\l,llsl + %ab\l4lsl -4b'llllsl -4abc^c^llllsl +2bc/,llsl -2bHlllsl + 4K^2^5 ^6 < - 2l',l',llsl + bY.sl - 2bc^l,llsl + I'Xsi - 2aWiy^ + 4ab\c^lisl -2b\l,llsl -4ab\l4lsl +2bY,lisl +2bY,iy^ - 2bc^l,llllsl -2bHlllsl +2bc^l,llllsl +bHtsl -a'b^lsl-na'b'slsl + 4a'b\c^slsl +Sa'b\l,slsl -4a'b\l,slsl +4ab'c^iyy^ - ea'b'llslsl -4ab\c^llslsl +4ab\llslsl -bY^s^ +2a'bY,slsl - 4ab\c^llslsl +Sab\l,l^slsl -2bY,l',slsl -bY^s^^ -2a'bY,slsl + 4ab\c^llslsl -Sab\l,llslsl +2bY,llslsl+2bH^l',slsl -bYy^s'^ + 2a'bY,slsl -4ab\c^iy^sl +4ab\ljyy^ -2bY,llslsl -2bY,llslsl + 2bYXslsl -bXslsl +Sa'b's;sl -4aV4 +Sa'b\l,sl -4aWiy^ + Sa'b'slsi -Sa'b\Lslsl +4aWsls' -4a'b'sis M3,=^aV.;< 128 M33 = 4a^bs^sl(2abc^c^ -labclc^ -b^s^ +2abc^c^sl -Ibc^l^sl -l^^ +1^5^ - ^n = -a'sl(4aW -^aWc^ -Aab'c^c^ +4ab'clc^ -Aabc^c^ll+Aabclc^ll + Aabc^c^ll -Aabclc^l] -Aabc^c^ll +Aabclc^l^ +Aabc^c^l^ -Aabclc^l^ + Ua'b'sl +b'sl -Sab\c^sl -Aa'bc^l.sl +Ab\l,sl -I6ab\l,sl - 2bXs'^ + Abc^lis'^ + ly^ - IbXs'^ + Sabc^c^l^sl - Abc^l^l^sl - 2^1^^ + Itsl +2bHlsl -Sabc^c^iy^ +%bc^l,llsl^2llllsl -2lliy^ +/^J -2b'l^sl + Sabc^c^l^sl^ -Sbc^lJ^sl -2lll^sl +2lll^sl -21^,1^^ +/,SJ -20a'bX + ■ 2Aab\l,sl -AbHls^ -AaWsl +AaWc'^sl -l6aWslsl+20aWslsl) M3, = -2as^s^{Aa'b'-ab\c^ -"^^'b^lcl -a'b\l, +AaWcll,+2a'bc^l, - 2ab'c^l^ - a\l', - 2abc^ll + ac^c^t^ - Aa^blj + 2ab^c^c^ll + a^c^l^l] + 2abc^l,ll-ac^c^ll+Aa'bll-2ab'c^c^ll-a\l,ll +b\l,ll -Aabc^lf, + c^llll^-2ac^c^llll -c^lfjl-ac^c^ll+c^yl -Aa'bl^ + 2ab\c^l^ + a\y^ -b\l,l^ + Aabc^l,l^ -c^l2l' -2ac^c^lll^ +c^l,lll^ +2ac^c^l^l^ - 2c^l,lie, -ac^c^l^+c^lj^ -Ga'b'sl +Aa'b\clsl -Sa'b\l,sl - Aa'bc^l^sl +Aab\l,sl +Aa^bl^sl +Aab\c^iy^ +6aX'^^ -^^bc^hll^l - ea'bllsl + Sabc^lJlsl - 2bl',iy^ + 6a'bl^sl - Sabc^l.l^sl + 2blll^sl - ea'b'sl +Aa'b'c,c^sl -AaWclLsi -2a'blhi +6a'blhl -6a'blhl + .2l3„2_2 , o„2i.2_ ; 2 1 T„2r;2„2„2 /:„2.t2 2 2 6a'biy^sl-6a'bl^slsl) M,, = a'b'sl -Aa'b'c^clsl +Aa%'cUl -Aa'bcJ^sl +Aa^b'cA,sl +2a'b'cjy. - '^^'h"^ '<h ^ (Ji'i"^ '(«l*2''^ '«>!'-2''0, Sa'b'cll.sl +aY,sl +2aWl',sl +Aa'bc^c^l',sl + Aa' befell', s^ -Aa^bc^l^sl 2a\llsl ^aY^sl -2aWllsl ^Aa^bc^cHlsl -Aa'bc^l.iy^ -2a\l,iy^ + 2aY,iy^ +aY,sl +2aWllsl -Aa'bc^cHlsl +^a'bc^l,iy^ +2aV^/,/^J - 129 Sa^bc^lJ^sl -la'c^lj^sl +2ab^c^y^sl +4a^l^l^sl +4abc^c^l^l^sl - 2ac^llljsl + 20^/4/7 5J -2ac^l^lll'^sl -2a^l^lTsl +4ac^l^lll'^sl^ -211iIItsI^ + aXsl -2acJ,i;sl +l',l^sl -4a'b'cy^ +Sa'b'cll,sl -Aa'bHy^ +'^a'b'sl + a'b'sl -4a'b\c^sl +2a'b'iy^ -4a'bc^c^l',sl +aY,sl -2a'bHlsl + 4a'bc^c^llsl -2a'llllsl +aY,sl +2aViy^ -4a'bc^c^iy^ +2a'l',iy - 2aHliy^ + aXsi - 2aWl^sl + 4a'bc^c^l^sl - 2a'l',l^sl + 2aHll^sl - 2aHll^sl +a'lUl -^a'b'slsl -a'b's^sl +4a'bc^l,slsl -4a'b\l,slsl - ea'b'iyy^ +4a'bc^iyy^ -aYyy^+2a'b'iyy^ +4a'bc^i,iy^si - 2aHiiyy^ -aYyy^ -2a'b'iyy^ -sa'bc^i,iy^si +2aY,iyy^ + 2aHliyy^ -aYyy^ +2a'bY,slsl +Sa'bc^l,l^slsl -2aY,l^slsl - 2aYjyy^ +2aHll^slsl -a'l^sX +4aWiy^sl -4a%'sX +4a'b'sX APPENDIX B CONSTANTS FOR THE FORWARD KINEMATIC ANALYSIS OF THE REDUNDANT 8-8 IN-PARALLEL MANIPULATOR g^=}-(-a^+ la" a - 2aW -b^ - 2b^J3^ + ll] - odl - ll + cdl +1^) q, = i (- a" + 5a^a - SaW + 4a V - 2b^/3^ + Aocb^ p^ + /,' - < + ll - 3odl ) 2(1 - 2a) a, = ^ (fe' - 2o(b^ -4b^l3 + Sccb^P - AaWp + ' 2b{\-2p + 2aP){\-2a-2l3 + 2apy Ab^p^ - 8c*'/?' + 4ar'& V + /,' - < - 2yffl,' + 3ay9,' - a^pl^ - ll + odl + 2/3?' - ^apll + a^pll + odl - apll + a^pll - apl] + a^pll - afill + a^pll + apll - a^fll + apll - a^pll - odl + apll - ^^fis ) q, = ? i2b^-4ab^-Sb^p + l6ab^p-Sa^b^P + ^ 2b{\-2p + 2aP){\-2a-2p + 2apy 8b^p^ - 1 6ab^p^ + Sa^b^p^ + if - aHf - 2pl^ + 2apll - ll + a^l -h 2pll - 2apll + odl + aHl - 2apLl + 2a^pll + all - a^l - 2apll + 2a^pll - ll + 2odl - a^l + 2pll - eapll + Aa^pil + 1] - 2odl + a^ll - 2pll + 6apll - Aa^pll - all + a^ll + 2apll-2a^pll-odl-aHl+2apll-2a^pll) ?6 -(-2b'^ +9oi}' - 2b{\ - a){\ -2a + 2ap){\ -2p + 2ap){\ -2a-2p + 2aP) UaV + 4a'b^ + 8b^p - AOab^p + 66aVp - AOa'b^p + Sa'b^p - 8b^p^ + 52ab^P^-96a^b^p^+68a^b^P^-\6a%^P^-\6od)^p'+40a^b^P^- 32a'b^p' + Sa'b^P' - ll + 3< - 2aHl + 3pll - 1 lapll +\3a^pll - la'pll + 2a'pil - 2p'll + SaP'll - IQa'P'll + 6a'P'll - 2a'P'll + odl - 3a'll + 2aHl + 130 131 la^pHl - aHl + 2aHl - pll + apll + a^pil - 5a' pll + la'pll + 2J3^l^ - 4aJ3^l^ + Qi 1 ■{p^-Acd)''+Aa'^b 2 1.2 2b{\ - a){\ -la + 2ap){\ -2p + 2ap){\ -2a-2p + 2aJ3) Ab^p + 1 %ab^p - 24a^b^p + Sa'b^fi + Ab^p^ - 2Aab^p^ + 36a^b^p^ - \6a'b^p^ + Sab^P' -Xea^b^P' + Uc'b^p' + al^ - 3a^l^ + 2a'l^ - 2apif + 6a^pl^ - Aa'pl^ + 2apH^ - ea^pH^ + Aa'pH^ - all + 3aHl - 2a'll - Aa^pl^ + Aa'pll + 2apHl - 2a^pHl + ll - Soil + 9aHl - 6a' l^ - 2 pi] + \Aapil - 30a'/?/' + 2^a'pll - %a'pll - 6apHl +\Za^pHl - 20a' pH^ + Sa^p^l^ - 1] + Aal^ - SaH] + 2a'll + 2pll - \Oapll +\6a^pll - Sa'pl^ + 2apHl - 6a^pY, + Aa'pH^ + all - ^oc^^l + 2a'll - 2apll + 6a^pll - Aa' pi] + 2apHl - 6a^P^ll + Aa'pH] + all - ^cc^^l + 2a'll - Aapll +\6a^pll - 2Qa'pil + ^a'pil + 2apHl - lOa^P^ll + \6a' p^l - Sa'P^ll - all + Sa^ll - 2a'll + 2apll - 6a^pll + Aa'pll - 2apHl + 6a^p^ll - Aa'pHl - aHl + 2a'll + 2apll - Aa^pil - 2apHl + 6a^pHl - Aa'pHl ) ?8 ■(^'-4o*'+4a'fo 2r2 2b{\ - a){\ -2a + 2ap){\ -2p + 2ap){\ -2a-2p + 2aP) Ab'p + \%od)^p - 2Aa^b^P + %a'b^ p + Ab^p' - 2Aab^p' + 36aW p^ - I6a'b^p^ + Sab'p' -\6aWp' + Sa'b^p' + a'lf - 3a'll + 2aUl - 2apll + Aa^pil - 2a' pll + 2apHl - 6a'P'll + Aa'p'll - a'll + 3a'll - 2aUl - 2a'pil + 2a'pll + 2apHl - 132 4aHl-2a) Aa\\-2a){b-0dj-q,-lq,+20(q,) 9 2 3a^/,^ + /j - 40/3' + 5a^l^ - lo'l] + odl - 3a^l^ + 2ar'/6 + 1^ - lodj + a'^lj + 2bq^ - %0d?q^ + IQa'^bq^ - Aa^bq^ - 4bj3q^ + llocbj^q^ - Sa^b/Jq^ - la^q] + Aa'ql - 6abfiq^ + I4a^bfiq^ - Sa^bfiq^ + 2aq^q^ - Aa^q^q^ - 4bq^ + l2oi)q^ - lOa^bq^ + Aa^bq^ + Safcyfi^^ ~ 24«^&/5^6 + lea^bfiq^, + Aq^q^^ -l2oij^qf, + Sa^qsqf, + 2ql - Saqj + lOa^qj - Ao^q] + 20*^7 - 6a^bq^ + Aa^bq^ + 2b pq^ - \2(Xbpq^ + 22a^bpq^ - I2a^bfy^ + 2q^q^ - 6oaj^q^ + Aa^q^q^ + 2bq^ - Aabq^ + 2bpq^ - Sabfiq^ + lAa^bfiq^ - Sa'bfiq^ - 2q,q^ + 6aq,q^ - Aa^q^q^ ) 9 -^ A^, , = -^— (a^b ~Aa^ab + Sa^a^b - 2a^a'b + bl^ - 2ablf + a'^blf - 2otbj3l^ + 2a^bj3lf + 2abpi^ - 6a'bfi^ + Aa'bfil^ - 2abl3ll + 6a^bpll - Aa'bpll - bll + 2(xbll - a^bll + 2ocbl3ll - 2a^bpll + b\, - 3o(b\, + 2aWq^ + 2b^ pq^ - eocb^pq^ + AaWpq^ + I'^q^ - 2odfq^ + a^lfq^ - 2l^q^ + lod^q^ - la^lq^ + 2a'llq^ + llq, - Aodlq, + Sa^lq, - 2aHlq, - od^q, + aHjq^ - 2bql + 6odjql - 133 4a^bql + 4abPql - Sa^bfiq] - la^oeq^ + Sa^a^q^ - lOa^a^q^ + 4aWq^ + lodj^Pq, - 2aWj3q, + Ib^p^q, - lOob^fi^q, + SaWfi\ - 2lfq, + Aod^q, - la^l^q^ + lodlq^ - 2aHlq^ + Ibq^q^ - 6ccbq^q^ + 4a^bq^q^ - Ibpq^q^ + 4cxb/3q^qs + 2a^b/3q^q^ - 4(Xbfiq^^ + 4a^bj3ql - 4a^q^ + ISa^O^g - ?>Qa^a^q^ + 24a'a'q^ - SaWq^ - 2b'' q^ + 5(Xb'q^ - 2aWq^ + 4ody'fiq, - Sa'b'/3q^ - 4b'/3\ + i2c(b'P\ -^a'b'j3\ -od^q,+a'l^q,+2l^q,-6od^q,+5a'l^q, - 2a'l^q, + Cdlq, - 3aHlq^ + 2a^llq, + 2llq^ - 6odlq^ + 5aHlq^ - 2bq,q^ + 6abq,q, - 4a'bq,q, - 4ocbfiq^q, + Sa'bj3q^q, + 2ab/3q,q^ - 6a'bj3q,q, + 2aq,q,q, - 4a'q,q,q, - 2bql + 40!bql + 4cxbfiql - Sa'bpql + 2q,ql - 6aq,ql + 4a'q^ql +a'q-j -Sa^oq-j +I0a'a'qj -lOa'a^q^ +4a'a*q^ -4od?'fiq^ + na'b'pq^ -Za^b'pq,+2b'p\, -2od)'p'q, -Sa^b^fi^q, +%a^b'p\, - l^q^ + 'iodlq-^ - 2a^/f ^7 - llq^ + liCdlq^ - 2a'l^q-, + 2bq^q^ - Sabq^q-j + lOa'bq^q-j - 4a^bq^q.j - 6b/3q^q^ + 22od}fy^q^ - 22a'bPq^q^ + 4a^bl3q^q^ - 2aqlq, + 4a'qlq^ - 2(Xb/Jq,q^ + 4a'bpq,q, + 294^5^7 - 4aq,q^q, + 2abq,q, - 6a'bq^q-j +4a^bq^q.j +4bfy^q^ -i^odyPq^^q.^ +lSa'bfy^q^ -4a^b/3q(,q^ + 2^496<?7 -6«?W697 +4a'q^q^qT+a'q^-5a^aq^+\0a^a'q^-l0a'a^q^ + 4a'a'q^ + b\^ - 2ab'q, - 2b^j3q^ + 6a'b'pq^ + 2b'fi'q^ - 2ab'fi'q^ + 2odfq, - 2a'lfq, - 2l^q, + 5od^q, - 2a'l^q, + 1^ - ^ocl^^s + 2a'llq, - l^q, + 2od^q, - 2aHlq^ + 2bq^q^ - 6abq^q^ + 4a'bq^q^ - 2bj3q^q^ + 6abJ3q^q^ - 6a'b/3q^q^ - 2aqlq, + 4a'qlq, + 2bfiq,q, + 4bpq,q^ -\6o(b/3q,q, +\M'bpq^q^ - 2q,q,q, + 6«S?4^698 - ^0C^<l4<l6^» + 2bq^q^ - 6oi>q^q^ + 4a^bq^q^ - 6bpq^q^ + 2QiOi>pq^q^ - I6a'bj3q^qs + 2bql - 4cxbql - 2bj3q^ + 4(Xbj3q^ - Aa'bpq] ) yV,o = a'(- a' + 4a V - 5a'ar' + 2a' a" - a'b' + 3a'Q*' - 2a'a^b' - 3a'lf + Sa'od^ - laWlf + 2aWl^ -b'l^ +cxb'l^ - 2b^J3'l^ + 2ab'/3'lf + 2aHl -9a'od^ + XSa^a^ll -Ua'a'll +4a'a'll +2b' PHI -eceb' PHI +4aWj3'l', +cdfl^ - aH'^ll - aHl + 6a'odl -UaWll + \2a'a'll - 4a'a'll - 2b'pY, + 6ab'/3'l', - 4a'b'pHl + l^ll - 2(xllll + aH^ll + 2a'll - 5a'od', + 5aWl^ - 2aWll + bY, - 134 ccbHl + Ib^pHl - lab'fiX + IX - cdfls - lllll + 50^3 ^8 - 3a'/3 ^8 + ^l^l - Aodlll + 3aHlll - /; + Odt - Aa^bpq, + 1 la^dbpq, - SaWb/5q, - 2b^ pq, + Aoi)'l3q, - IbplU, + 2od)pl^q, + labpllq, - Aa'bpllq, + Ibpllq, - 6c(bj3l^,q, + Aa^bpllq, + locbpllq^ + Aa^ql -\6a^aql + lOaWql - SaWql + Ib^q] - 4ceb'ql - lllql + Aodlq] - lllq] + Aodlql - 2a'b/3q, + 4a^ab/3q, - 6aWb/ii, + 4aWbPq, - lodj'pq, - 4b'/3'q, + 4ab'fi'q, + 2bj3lfq, - 2cd?j3l^q, + 2a^bfil^q, + 2(Xbpllq^ - 2a^bl3llq^ + 2bpllq^ - Aocbfil^q^ - 2a^q^q^ + lOa^oq^q^ - \6a^a^q^q^ + SaWq^q,-2b^q^q, +4(Xb^q,q, -4od)^fi^q,q,+4l^q,q,-l0odlq^q,+4aH^q,q, - 2llq,q,+6odlq,q, -4aHlq,q, + 2llq,q, -4cdlq,q, +4bfiqlq, -Sabfiqlq, + 4ab^P^ql -4bpq^ql+%0i>Pq^ql + 2a^bq^ -\0a^Ocbq(,+\6aWbq^ -Sa^a^bq^ + 4b'fi\ - Sab'fi'q, - 2blU, + 2od>l^q, - 2bl',q, + 6otbl',q, - 4b'j3q,q, + Sod?'^,q, - 4odj^fiq,q, + 4bq,q,q, - 8(Xbq,q,q, + 4a^ql -IGaWe + 20aWql - SaWql + 2b' ql - 4(Xb'ql + 4b'fi'ql - Sc(b'/3'ql - 2l^ql + 4od^ql - 21^ ql + 4cd^ql -2a'bq.j +\2a'od?q^ -26a^a'bq.j +24a'a^bq^ -Sa'a'^bq^ +4a'b/ij^ - \8a'ocb/3qT +30aWbJ3qT -24aWbfiq^ +Sa'a'bj3q^ -4b'fi'q,+\2(xb^J3'q, - Sa'b'p'q^ +4b^fi\,-\2ab^p'q, +8a'b^j3^q, +2bl^q, -4o(bl^q, +2a'blfq^ + 2abfl^q^ - 2a'bpl^q^ + 2bllq, - %oi>llq, + 6a'bl^q^ - 4bj3l^q^ + lOoibpl^q^ - ea'bpilq, ■\-4b'pq^q,-\2(Xb'pq,q, +8aWfiq,q, +4ocb'[5\,q, -8a'b'/3'q,q^ - 4bpqlq, +^CCbpqlq,+4cd)''l3q,q, -4a'b'Pq,q, +4a^b'fi'q,q, -4bq,q,q, + l2oi>q^q^q^ -Sa'bq^q^q^ +Sb/3q^q^q^ -20oi>fiq^q^q-j +8a'bfiq^q^q.j -2a'qf^q^ + Wa'aq^q^-ieaWq^q^ +Sa'a'q^q, -4b' P'q^q^ +'&ccb' P\^q, +2l^q^q^ - 2cdU(,Qi + 2/8^697 -^cdUt^li+^bpq^q^q^ -8o±>Pq^q^q^ +4od?fiq^q^q^ - 4^4^5^697 +^Otq,q,q^q, -4bpqlq, +8abfiqlq, -2a'bq,+\0a'abq^-l6aW^^^ . Sa'a^bq^ + 4a'bpq^ -XQa'ocbpq^ +\0a'a'b/3q^ -4a'a^bPq^ + 2b^ Pq^ - 2ab^ pq^ - 4b^p'q, + Sab'P'q^ + 4b'p'q, - 4od?^p'q^ + 2bl^q, - 2c(bl^q, + 2bpi^q, - 2cxbpl'q^ - 4bpl^q^ + lOabpl'q^ - 6a'bpllq^ + 2bpl^q^ - Socbfil^q^ + Ga^b/Jl^q^ + 2bllq, - eccbllq, - 4bpilq, + 4(xbpilq, + 4b' fiq.q, - 8o(b'j3q,q, + 4od?'j3'q,q, - 4bpqlq^ + Sotbfiqlq, + 4ab'j3q,q, + 4b'fi'q,q, - 8ab'fi'q,q, - 4bq,q,q, + 135 Sabq^q^q^ + ^bpq^q^q^ - Sobpq^q.q^ - la^q^q^ + Wa^Oq^qg -\6a^a^q^q^ + SaWq.q, - 2b^q,q, + 4od?'q,q, - 4b^fiq,q, + \2ab^/3q,q, - 4b^/3\q, + Sab^fi'q.q, + 4l^q,q, - \0(Xl^q,q, + ^aH^q^q^ - 2llq,q^ + 6cdlq,q^ - Aa^lq^q^ 2^8 <7698 - 4crf8 96^8 + ^bpq.q^q^ - Mbpq.q^q^ - 4bfiq,q^q^ + %od}pq^q^q^ + Abqlq, - Socbqlq, - 4bfylq, + ^obfiqlq, - 4a^aq^q, +\2aWq,q, - SaWq.q, - 4b'j3q,q,-\6(Xb'j3q,q, + l2a'b'j3q,q,-4b'fi'q,q,+l2ab'fi'q,q, - Ua^b^fi^q^q^ -4lfq^q^+4odlq^q^+4cdlq,q^ -Sbpq.q^q^ +\6ab/3q,q^q^ + 4qlqiqi -^0(qlqiqi-'^Oi>lk5Qi(i%-^bq^(ii^^'^\2oi)q^q,q^ -%a^bq^q^q^ + l2bfiq,q,q,-32c(bfiq,q,q,+Sa'b/3q,q,q,+4b'fiql-\2odj'fiq', -4b'fi'q', + 40(b'P\l - 4bq,ql + Mbq.ql + 4bPq,ql - Sabfiq.ql + S(Xbj3q,q', ) + N..= 8fl^a(l - 2a) ^23 = -^^-^—^{b -oi)- 4abJ3 + 4a^bp + 20" q^ +aq,+ 2aq^ - 2a^q^ +q,-aqT + (I ~ OC) N,, = ^^(-2a^ +16a^a-46aV +60flV -36aV +8aV +&' -5o*^ + 8a'&' - 4a'b^ - 4b^fi^ + 16c*'/?' - 2QaVp^ + 8a'&'y9' + /,' - 3< + 4a'/,' - 2a'/' + 2/3 - 8crf' + lOa'/' - 4a'/' - ll + 5cdl - %aHl + 4a'/,' + 2/g' - \Qodl + Ha'/g' - 6a'/g' - 66^4 + 24c*^4 - 30a'^94 + Ua^bq^ + 4b pq^ - %(xbpq^ - 4a^bpq^ + Sa^bfiq^ - 2bq^ + Sodjq^ - lOa^bq^ + 4a^bq^ - 2b/3q^ + \4odj/3q^ - 24a^bfiq^ + \2a^bfiq^ - 2a^q^q^ + 4a'94^5 + 2/7^6 ~ lOofe^g + Ua'/?^, - 4a^bq(, - locq^q^ + 6a'^5^6 - 4a^q^qf^ + 6oi)q^ - 1 Sa^bq^ + 1 2a^bq^ + 4bPq^ - 20cdj/]q.j + 32a^bfiq^ - \6a^bPq^ -6aq^q-i +\Sa^q^q., -Ua^q^q^ -2q^q-, +6c(q^q-, -4a^q^q^ -2qf^q.^ + Saq^q^ - lOa^q^q^ + 4a^qf^q^ - 6bq^ + 22oi>qg - 22a^bq^ + 4ar'bq^ + 4b Pq^ - \6cd)pq^ + ^6a^b/3q^ - 4a^b/3q^ - 2aq^q^ + 6a^q^q^ - 4a^q^q^ - 2q^q^ + Saq^q^ - \0a^q^q^+4a^q^q,) T^r^' 136 N., = la' '' \-a [a'b -4a^ab + 5aWb - la^a'b + 2b' - 6ab' + 4a^b' - Aa'abp + l6aWbj3 - lOaWbfi + SaWbfi + 2bl^ - 4abl^ + la^blf - 2oi>pl^ + la^bpl^ - ml + \Oabll - 9a^bll + la'bll + lod^pll - ba^bpll + Aa^bpl] + bll - AMI + 5a^bll - la'bll - Icdjpll + 6a'bJ3ll - Aa'bpll - IMl + la^bll + loebpll - la^bl3ll-2a^q^ + \0a^Ocq^-\%a^a^q^ + \Aa^a\^-Aa^a\^-3b^q^ +9ab^q^ - ea'b'q, + 2b'fiq, - 60d?'/3q, + Aa'b'fiq, - Ab'fi'q, + Uocb'fi'q, - Ba'b'fi'q, + 3llq, - 9odlq, + 6aHlq, + llq, - Sod^q, + 2aHlq, - a\, + 6a^0Cq, - l3aWq, + Ua'a'q, - Aa'a'q, - b\ + 3(Xb'q, - 2a'b'q, + 2b'fiq, - 2odj'fiq, - 2b'fi'q, + 2ab^P\^+2llq5-l0dlq^+la^llq^ -2aHlq^ -llq,+Aodlq, -Sa^l^q^ + 2aHlq, + llq, - 3odlq, + 2aHlq, - Abq.q, + \2(Xbq,q, - Sa^bq.q, + 2bfiq,q, - 2c(bpq^q, - Aa^bpq^q, - 2b pq] + 6odjpql - Aa^bpq] + 2a^(Xq^ - Sa^a^q^ + lOfl V^e - 4a^«'96 - 2^^^6 + 5a^ ^6 " 2a^b^q^ + Aab^ pq^ - %aW pq^ + od^q^ - 2aHlq^ -Odlq, + 2aHlq, + 2bq,q, - 6abq,q, + Aa^bq.q, + 2a'q, -9a'aq, + ISa^a^q^ - \2aWq^ + Aa^a^q, + 3b^q^ - 9oi)^q^ + ^Wq^ - Aa'b^q^ - 6b^pq^ + leab^pq^ - 6a'b^pq^ - Aa'b^flq^ + 2b^P^q, - 2ab^P^q, - Sa^b^P^q^ +Sa'b^P^qT+2odfqT-2a^l^q^ -l^q^ +5aHlqT -2a'llq^ +od.lqn - Sa^l^q^ + 2aHlq, -llq, + 3odlq^ -Aa^lq^ + ^bPq.q, - \6abpq,q, + Sa^bpq^q-j - Ibq^q^ + Sobq^q-j - lOa^bq^q^ + Aa'bq^q^ + Abpq^q^ - \2od)pq^q^ + lAa^bPq^q^ -Aa'bpq.q, -Aaq.q^q^ +Sa^q^q^qT +2bq^q^ Sotbq^q^ + Sa^bq.q^ - 20CbPq^q^ + Aa^bpq^q-, - 2q,q^q, + 6aq,q,q^ - Aa^q^q^q^ + 2abq^ - 6a^bq^ + Aa'bqj - 2bpq^ + Aoi>pql + 2a^bpq^ - Aa'bpq^ - 2q^q] + 6aq^qj - Aa^q^q^ +a^C(q^-5a^a^q^+%a^a'q^ -Aa^a^q^ -cd?^q^ + 2a^b^q^ -Sod^^pq^ + UaWPqs - 2b^p^q^ + lOob^P^q^ - Sa^b^p^q^ + 2llq^ - 6odlq^ + 5a'^llq^ - 2a'llq^ + Cdlq^ - 3aHlq^ + 2aHlq^ - OCl^q, + 2aHlq, + 2bpq,q, - ^Obpq^q^ + Sa^bPq.q^ - 2bq^q^ + 6od)q^q^ - Aa^bq^q^ + 2bpq^q^ - Aobpq^q^ - Abq^q^ + lOabq^q^ - Aa^bq^q^ + 20!bPq^q^ - Aa^bpq^q^ + 2bq^q^ - 2abq^q^ - 6a^bq^q^ + Aa'bq-^qg + 2bpq^q^ - lOoirpq^q^ + lOa^bPq^q^ - Aa'bPq^q^ - 2q^q^q^ + ^Oiq.qjq, - Aa^q.q^q, + 2abql - Aa^bql - 2ocbPql + Aa^bpql ) 137 iVjo = a'(-2a' + 14aV-38flV +50flV -32aV +8flV -aV +3a'a*' - 2aWb' -b' + 2c(b' -Aa'b'fi' +I6a'ci>'fi' -20aWb'fi' +SaWb'fi' - 2b'lf + 2(xb'l^ - 2b'fi'lf + 2cd>^fi^lf + 3aHl -Ua^odl + 2Qa^aHl - lAa^aH] + Aa'a'll + 3b^ll - lotb'l^ + 2aWl^ + 2b'fi'l^ - 6(xb'fi'l', + Aa'b'p'll + 2lfl^ - 2cd^ll - 2ll + 5< - 2aHt - 2aHl + IQa'all -l^a'aHl + lAa'a'll - Aa'aY, - b'll + 3(xbY, - 2a'b'll - 2b'l5Hl + 6(xb'fi'll - 4a'b'/3'l', + l^l - 3od^l', + 2aHlll + ZaHl -\3a'od^ + ISaWl', - SaWl^ + 2od?X + ^b'fi'l^ - 2ab'P'l', - lUl + odlll - 2aHlll + llll - 3odlll + 2aHlll - it + 2< + Aa'bPq, - \6a^(Xbpq, + 20aWb/3q, - SaWbj3q, - 2b'j3q, + Aab' pq, + 2bpllq, - Aab/3l^q,-2bfil^q,+Aabj3l^q,-Aa^bj3q, + l6a^odjfiq,-20a'a^b^,+ %a^a'bPq, - 2b' Pq, - Ab'P'q, + Aab'P'q, + 2bpllq, + 2bpilq, - Acxbfil^q, + Ab'q.q, - %oi>\,q^ + Ab^ fi^^q, - Sab^p\,q, - Al^q.q, + Sod^q.q, + Aa^bq, - 16a^abq^ + 20aWbqf, - Sa^a'bq^ + 2b' q, - Aob'q^ + Ab'fi^q^ - Sab'fi^q^ - 2bllqf^ + Accbllq^ - 2bl^q^ + Accbl^q^ - Aa^bq^ + 2Qa^od)q^ - 36aWbqj + 2SaWbqy -Sa^a*bq^ -2b'qT + 6ccb'q^ -AaVq^ ^ea^bpq^ -2(ia^(Xbl3q^ + A0aWb/3q^ - 2Sa^a'bfiq, + Sa^a'b/3q^ + 6b'fiq^ - lAodj'fiqj + Aa^b'fiq, - Ab'P'q,+l2c(b'fi^q,-^a^b'fi'q,+Ab'fi'qT-\20Cb'l3'q,+Sa'b'fi'qT + Abplfq^ - Aabpllq, + 2bllq^ - 6ccbllq, + Aa^bl^q^ - Wpilq, + 20abpi^q^ - %a^bpllq^ + 2bpllq^ - 6(Xbpllq^ + Aa^bfil^q^ + 2bl^q^ - 6ocblgq^ + Aa^bl^q^ - 2bj3l^q^ + 2od)pllq, - Aa^bpllq, + Ab^j3^q,q^ - %ab^p^q^q^ + Ab^/3^q,q^ - Wpq,q,q^ + I6abfiq,q,q, - Aa'q^q^ + I6a^aq^q^ - 20a^a^q^q^ + Sa^a'q^q^ - 2b\^q, + Adb\s, - Ab^Pq^q^ + Sob^fiq.q^ - Ab^fi'q.q, + &od?'fi\q, + 2/3^6^7 - 4crf'9697 + ^^ke^ii - ^odkeQi + 2fl'^7 " lOa'os?' + 16a'a'^7 - Sa^a'q^+Ab^/3q^-l2ab^/3q^+Sa^b^/3q^j -Ab^jS^q^^ +l2od?^j3^q^ - Sa^b^fi^q^ - 2lfq^ + 2cdfq^ - 2l^q^ + 6od^q^ - AbJ3q,q^ + 2>(XbPq^q'^ - AC(bj3q,q^ +Aq,q,q'^ -Soq.q.q^ + Abj3q,q^ -^(xb/3q,q^ -Aa^bq,+l6a^od?q, - 20a^a^bqg + Sa^a^bq^ - 2b' q^ + Aoi}'q^ + 6a^bpq^ - 26a^od}pq^ + 36aWbfiq^ - lea'a'bfiq^ + Aab'fiq^ - Ab'fi^q, + Socb'fi^q, + Ab'fi'q^ - 138 2 _ eod^pilq, + Aa^bpilq, + Ibllq, - 4abl^q, - Abpijq, + Sod?j3l^q, - 4b' P\, SO(b'/3'q,q,+4b'P'q,q, -Sccb'fi'q,q,+4b\q, Sob'q.q, -4b'/3q,q, + ^ab'j3q,q,+4b'fi\q, -Sab'fi\q, -4l^q,q,+Sod^q,q,-2a'q^q, + XQa'aq^q^ - Iba'a'q^q^ + SaWq^q^ - Ib'^q^q^ + 4ab\^q^ + Sb'fiq^q^ - lOab'pq^q, +Sa'b^fiq,q, -Sb'j3'q,q,+l2c(b'P'q,q, -Sa'b'j3\^q,+4l^q,q, - lOod^q.q, + 4aHlq,q^ - 2l^q,q, + 6cxl^q^q, - 4a'llq^q, + lljq.q^ - 4od^q,q, + 4bpq,q,q^ - Sab/3q^qjq, - 4bPq^q,q^ + %cA)fiq^q,q^ + 4bq^q,q^ - 2>abq^q^q^ - ^bpq(,q-,q^ + Sab/3q,q^q, - 4bq^q, + llocbq^q^ - ^a'bq^q, + 4bPq^q, - Sobfiq'q, + Sa'bflq'.q, + 4b'/3ql - Sc(b'fiq', - 4b'[3\l + ^0(b' P\l - 4bq,ql + m?q,ql + .,0 _2 o„.i.fl_ _2\ _ 8a'ar(l-2a) iV^^ = _ 4a (1 2a) ^^ -3c(b + 2a'b - 4abp + 4a'bJ3 - 2a\, + 3aq, + 20^^ - 2a\, + q^-aq^+q^-aq^) ^ 2a\\-2oO ui _ j2^2^ ^ 22a'«' - 16a'a^ + 4fl V + &' - 3£*' + la'b" + '' (1-a)' 4fe'y9' - 8c*'y9' + 4a'&'y9' - 1] + ai,' - 2/3' + 60^3' - (id'l] + 2a'/3' + ll - Zcdl + 4a''ll - la^l - 2ll + 4cdl - la^l - 2bq^ + 4abq^ - la'bq^ - 4bj3q^ + I6(xb/3q^ - 1 2a'b/3q^ + 4bq^ - Sabq^ + 4a' bq^ + 2b Pq^ - 1 %dbpq^ + 1 6a'b/5q^ -2a'q^q^ + 2aql - 26^6 + 2c*9g - 2a'bq^ + 6aq^q^ - Ga'q^q^ + 2bq^ - 6od)q-, + Sa'bq^ - 4a^bq.j -4bPq^ + 8abfiq.j - Sa'bfiq^ + 4a^bpq^ - 2aq^q^ + 2a'q^q-, + 4q^q^ - 4aq,q, + 2q^q, - 4aq,q, + 2a'q,qy + 2abq, - 4bfiq, + Uabfiq, - Sa'b/3q^ - 6aq,q^ + 6a'q^q^ + 2q^q^ - 2aq^q^ + 2q^q^ - 4aq^q^ + 2a'q^q^ ) 139 w , = ^^(a'^ - 6a'fl* + na^a^b - Ua'a'b + 4aWb - Aa'obfi + Xea'a^bp - 20a^a'bp + SaWbfi - blf + 2ablf - a^bl^ + labfil^ - la^bpL^ - loiyfil + ea^bpll - Aa'bpll + Kxbpil - Ga'bffll + Aa'bfil^ + blj - loi)ll + a^bll - locbpll + la'^bpll + 20^4 - 10a'a94 + \SaWq, - Ha^a'q, + Aa'a'q^ + b^q^ 3(xb\ + 2a'b\-2b'fiq,+6c(b'Pq, -4a'b'j3q,+4b'fi\-l2ab'fi\ + SaW/3\, - l^q, + 3odlq, - 2aHlq, - 3llq, + 9(xllq, - 6aH^q, - 3a'q, + ISa'o^s -39aWq^+36aWq, -na^a'q,-2b\^ + 6ab^q^ -Aa^b'q^ - 5 + 2aWPq, - 6b^fi^q, + 22ab^j3\ -\6a^b^fi\ + ifq, - 2alfq, \-4l'; + 3, 22abfiq,q,- , ,_, ., ., \Sa^bfiql + 2a^aq^ -SaWq^+lOaWq^ -4a^a*q^ +ab^q^-2aWq^ - 4ab'/3q, + Sa'b^j3q, - al^q, + 2aH^q, + Od^q, - 2aHlq, + 4bq,q, - I2(xbq,q, + %a'^bq^qf,-2aqlq(^+4a^qlqf, +a^aq.j -Sa^a^q^ +Sa^a^q-j - 4a^ a* q^ + 4od}^ pq^ - \2aWl3q, +%a'b^pq, -2b^fi^q,+2ody^fi'q, +SaWp^q, -Sa'b^J3\^ -od^q, + aHlqT+2llq^ -Sodgq^ +3a^lsqT -2bfiq^q^ +4abpq^q^ -4bq^q^ +l6odjq^q^ - 20a^bq^q^ + Sa^bq^q^ + Sbfiq^q^ - 2Sody/3q^q^ + 26a^bpq^q^ - Sa^bfiq^q^ + ^ 2aq,q,q, -4a\,q^qT -2q]q, +4aqlq, +20(bl3q^q, -4a^bPq^q, -4q,q^q, + 12095^6^7 -Sa^q^q(,qj +2a^q^ -9a^(Xq^+\5a^a^q^-\2a^a^q^+4a^a*q^ + b'^q^ - 4oi)\, + 4a^b\^ + 2b^j3q, - 6a^b'j3q, + 2b^p^q, - lOab^fi^q, + SaWj3'q, - Odfq, + a^l^q, - l^q, + 6od^q, - lOa^.q, + 4a'l^,q, - 2odlq^ + eaHlq^ - 4aHlq^ - llq^ + 3odlq^ - aH^q, - 6bj3q^q, + 2Qoibpq^q^ - lea^bpq.q^ - 4bq,q^ +I2abq,q^ - Sa^bq.q, + 6b/3q^q, - 24c(bfiq,q^ + 20a^bfiq,q^ + 2aq^q,q^ - 4a^q^q^q^ + 2^96^8 " 2«*^6^8 " ^OC^bq^Qi " 2a*^6^8 + ^^C^bpq^q^ - 2q^q^q^ + ^O&is^i^s - 4^^959698 - 4«f'97^8 + i2a^bq^q^ - Sa^bq^q^ + 2bpq,q^ - 2od)pq,q^ - 6a^bpq,q^ + Sa'bfiq^q^ + 2q,q,q^ - 6c(q,q,q^ + 4a\^q^q^ - 2bql + 2oi}ql + 4a'bql - 2bPql + lOcd^Pql - lOa'bPq', + 2q,ql - 6aq,ql + 4a'q,ql ) 140 ^^, ^.^,+4aWbPq, + 2cd?'fiq,+4b'j3'q, . ,^ labpllq, - ^bpllq, + eabpllq^ - Aa\,q^ + \6a^aq,q^ - 20aWq^q, + Sa^a'q,q,-2b\,q,+4ab\q, -4b^/3\,q, +%(Xb^ p\,q,+lllq,q, - Acdlq^q, + 2llq,qs - ^cdlq^q, + 2a^ql - XQa^CXq] + \6a^a\l - %a^a'q] + 2b' ql - Aoib'ql + Ab'p'q] - Sod^'fiY, - AlWs + XOoilq] - Aa^Ws + 2llql - 6odlql + Aa'llq] - 21^, + Aod'.ql - AbPq.q] + Sobfiq.q', + Ab/Jql - ^ab^q] - Ab'P'q, + Socb'P\ + Abllq, - Sabl^q, - Abqlq, + ^ocbqlq, + 2a''bl3q, - l0a'od?fiqT+l6aWbPq^ -SaWbfiq, +Ab'p\^ -I2ceb'fi'q, +Sa'b'fi'q^ - Ab'fi'qj + \20Cb^P\, -%a'b^p\^ - 2bpllq^ + 2abfilfq, -Abl^q, + \2od)llq, - Sa'bl^q^ +6bl3l^qT -lAabfil^q^ +Sa'bj3l^q^ -Ab'' P^^q-^+^OCb' p'q^q^ - A0(b'/3\,qj+Abfiq,q,q^ -Sobpq.q.q^ +Abq',qT -I2(xbqjq, +Sa'bq',q^ - %bPq]q, + 20ctb/3qjq, - %a'bl3q]q, + Ab'fi'q.q, - Sccb'fi'q.q, - Al^q.q, + SCKfg 96^7 + Mke^li - ^OCqU(,Qi + ^a^b^q^ " ^Oa'obfiq, + \2aWb/3q, - AaWbJ3q, - 2b' Pq, + 2ab'pq, + Ab'fi\ - Sab'fi'q, - Ab'fi'q, + Aob'fi'q, - 2bj3lfq^ + 2oijpl^q^ + 6bpllq^ - lAob^^q^ + Sa^b/Sl^q^ - Abfil^q^ + l2ocbj3l^q^ - %a'bpilq, -Abllq,+^CCbllq,+Abpilq, -Aabfillq^+Ab' P\,q, -Sab'fi^q, - Sb'fi\q, + Uab'P'q.q, + Abj3q,q,q, - ScxbPq,q,q, + Abqlq^ - Sobqjq, - Abpqlq,+S(Xbpq',q, -Aa'q,q,+l6a'cxq,q,-20a'a'q,q,+%a'a'q,q, - 2b'q,q,+Acxb'q,q,+Sb'fiq,q,-l6ab'fiq,q, -Ab'/3'q,q,+Sab'j3'q,q, + 141 SaWq^q, -Sb'Pq,q,+24ocb'fiq,q,-l6a'b'j3q,q,+Sb'/3'q,q,-20ab'j3'q^q, ■ 4a\l-2a) , ,_ ... , . N,,=-^^-^^^^{b-(xb-2abfi + 2a'bfi + aq,+q,-aq,) (l-ay , . A^,, = ^^(2a'«-10aV +18aV -14aV +4aV -c*' +3aV -2a'&' - '' (1-a)'^ 2b^fi^ + Sotb^fi' - WaVfi' + 4a'b^/3' + if - 3cdf + AaHf - laHf + 1] - 4od^ + Sa^l - la'll + oil " ^X + 2a'll - lodj + AaX - laHl - 2^94 + ^oijq, - lOa^bq^ + 4a' bq^ + Ibpq^ - 6abPq, + 4a^bpq^ - a^q] + 2a' q] + 4abJ3q, - Sa^bPq, + 4a'bPq^ - 4bq, +\2abq, - lOa^bq^ + 4a' bq^, + 4abPq, -\2a^b/3q, + Sa'bfiq, - laq.q, + 6a\,q, - 4a'q,q, - ql + 4aql - 5a\l + la'q] + 2bq, - 4ocbq, - 2a^bq, + 4a'bq, + 2b pq, -\2oi}Pq, + 22a^b/3q, - \2a'b/3q, - 2q^q^ + 6as?4<?7 - 4a^q^q^ + 2dbq^ - 4a^bq^ - 4abfiq^ + Sa^bfiq^ - 4a'b/3q^ ) A^,, = -^ (a'b -4a'ab + 5aWb - 2aWb + b'- 3ab' + 2a^b' - 2a^abJ3 + \-a SaWbfi - lOa'a'b/3 + 4a'a'bfi + 2blf - 4ablf + 2a^blf - 2ab/3lf + 2a'bJ3lf - 2bll + loijll - la'^bll + 2a'bll + 2abl3ll - 6a^bpll + 4a'bpll + bll - 4abll + Sa^bll - 2a'bll - 2ab/3ll + 6a^bpll - 4a'bpll - bll + (^^l + 2(Xbl3ll - 2a^b/3ll 142 a^q,+6a^ceq^-\3aWq,+\2aWq,-4a^a*q,-b^q,+3odj^q,-2a^b^q,+ 2b' Pq, - eab'fiq, + 4a'b'fiq, - 2b'/3'q, + Gob'fl'q, - 4a'b'fi'q, + 2l^q, - lodlq, + laHlq, - 2aHlq, - llq, + Aodlq, - SaH^q, + 2aHlq, + l^q, - 30dlq, + 2aHlq, - 2bql + 6C(bql - Aa'bq] + 2b Pq] - Aoeb^q] + 2b' Pq] - 2ab'pq, - A0d)'P'q, + Aa'b'p'q, - 2bpq,q, + 6oijpq,q, - Aa'bpq.q, + a'aq^ - Sa'a'q^ + SaWq^ - 4aWq, - 4b'q, + 1 locb'q^ - 6a'b'q^ + Aab'Pq, - SaVPq, - 2b' P\ + 6ab'P\ - Aa'b'P'q, + 2l^q, - 6od^q, + Sa'llq, - 2aH^q, + Od^q, - Za'llq, + 2a' lU, - Odlq, + 2a'llq, + 2bq,q, - 6abq^q^ + Aa'bq.q^, - 2od)pq^q^ + 4a'bj3q,q^ + 2ocbj3q^q^ - Aa'bPq^q^ - 2bql + Aoi)ql + 2oi)pql - Aa'bpq] + a'q^ - 5a'aq^ + XQa'a'q^ -Wa'a'q^ + 4a V^7 + 3b' q^ -Wocb'q^ +\Q)a'b'q, - Aa'b'q, - Ab' pq, + Wab'pq^ - 2a'b'pq^ - Aa'b'pq, + 2b'P'q, - 2(Xb'p'q, - ^a'b'P'q, + Sa'b'p'q, + 20d^q, - 2a'lUn - 2l'qi + ^od'q^ - 2a'l^q, + l^q^ - 3odlq, + 2a'llq, - ijq^ + 2a/g ^7 - 2a'llq.j - 2bq^q^ + %odjq^q^ - XQa'bq^q^ + Aa'bq^q^ + 2bpq^q^ - 6od?Pq^q^ + ea'bpq^q^ - Aa'bpq^q^ - 2aqlq^ + Aa'qlq^ + 2bpq^q^ - Aodjpq.q, + Aa'bpq.q^ - 2a'bq^q, + Aa'bq^q^ + Abpq^q, - 1 6od)Pq^q^ + 1 %a'bpq^q^ - Aa'bpq^q^ - 2q,q^q^ + 6aq^q^q^ - Aa'q^q^q^ + 2bq^ - 6cdyq' + Aa'bq' - Abpq', + 1 2abpq' - Sa'bpq^ + 2b' q^ - 6odj'q, + Aa'b'q, - 2b' Pq, - 2(Xb'pq^ + Sa'b'Pq^ + Aodj'p'q^ - Aa'b'p'q^ - 2bq^q^ + 6ocbq^q^ - Aa'bq^q^ + 2bpq,q^ - 6abpq,qg + Aa'bpq^q, + 2abq,q, - Aa'bq^q^ - 2cd)Pq^q, + Aa'bpq^q^ + 2bq,q^ - Acxbq^q^ - Ibpq^q^ + Aocbpq^q^ - Aa'bpq.q^ ] N^ = a^(- a" + %a'a - 25a* a' + 38^ V' - 28fl V + Sa'a' - 2a'b' + 6a'ab' - Aa'a'b' -b' + 2ab' -Aa'b'p' +\6a'ab'p' -20a'a'b'P' +Sa'a'b'p' - Ab'l^ + Aab'lf - Ab'P'lf + Aab'p'l^ + 40^/3' - 1 Sa'od^ + 30a'a'l^ - 2Aa'aHl + %a'aHl+Ab'll - moi}'ll+Aa'b'll+Ab'p'll-\2od)'p'll+%a'b'p'll + Al^ll - Aod'l' - 4/' +120/3' - 9a'/' + 20^/3 - 2a' l^ + \2a'odl - 26a' a' ll + 2Aa'aHl - Sa'aW, - 2b'll + 6ab'll - Aa'b'll - Ab'p'lj + Uab'p'l^ - 143 %aWp'll + Aim - ^"^lll + ^^oc^ll - ^ocWe - ll + 4< - 5aX + 2a'C + la^l - XOa^odl + Xea'a^ - ^aWl^ + Ib^l + Ab'' fi^ - Aab'fi'l^ - Al^ + eodlll - Aa'l^ + 21^1^ - eodUl + AaHlll - 1* + 2< + Aa'bPq, - 20a'od?/3q, + 32aWb^,-l6aWbfiq, -Ab'fiq,+S(xb'j3q,+Aab/3l',q, -Sa'bfil^q, + Abpllq^ - ncdj/Mlq, + M^bjMlq, - Abfil^q, + Sobfy, + Ab^ql - 80*^^4' - 4/3^^4' + Sod^ql - Aa'bfiq, + 20a^abPq, - 32aWb/3q, + I6a'a'bfiq, - Ab'j3q, - Sb'fi'qs + Sab'fi'q, + Sbfil^q^ - I2abj3l^q, + Sa^b/3l^q, - Abpllq, + nocbpllq, -Sa'bfillq,+Abfil^q,-^abj3l^q,+Sb^P'q,q,-l6ab^P^q,q, - Ab^fi^ql + Sab^fi^l + Aa'bq^ - 20a^abq^ + 32aWbq^ - \6a^a^bq^ + Ab^q^ - Sab^q^ + Sb^P^qf, - \6ab'fi^q^ - Sbl^q^ + 20abl^q, - Sa^bl^q^ + Abl^q^ - nabl^q, + Sa'bllq, - Abllq, + %CXbllq, - Sb'j3q,q, + l6(Xb'j3q,q, + Sb'/3q,q, - \6ccb^pq,q,+Ab^P\l -Sob'fiY, -Al^ql +Sod^ql -Aa^bq, +2Aa^abq, - 52aWbq.j + ASa^a^bqy -l6aWbq^ -Ab^q.+Uob^q^ -Sa^b^q^ +Sa^b/3q^ - 36a^abJ3qT+60a^a^bfiq, -ASaWb/^q^ +I6a^a'b/3q, +Sb^/3q^ -20ab^J3q^ + Sa^b'fiq, -Sb'fi^q^ +2Aab'fi^qT -lea'b'fi^q^ + %b' P\, - 2Aocb^ p\, + lea^b'fi'q^+Sbfilfq^ -Sotbfil^q, +Sbl^qT -2^abl^qT +2Sa^bl^q^ -Sa'bl^q^ - \6bpllq,+A%oi}pllq, -36a^b/3l^q, +Sa^b/3l^q., -Abl^q^ +I6abllq^ - 20arX^7 +^oc^bllq, +%b/3l^qT -2Socb/3llqy +2^^bfy^ -Sa^b/M^q^ + Abllq, -nodjllq, +2,a^bllq^ -%bpilq,+\2cd)pllq^ -%a^bpllq,+%b^pq^q^ - 2A(Xb^j3q,qj+l6a^b^/3q,q, +Sab^j3^q,q,-I6a^b^fi^q,qj -Sbfiqlq^ + Itabpqlq^ -W^ pq.q^ +2A0i}^ pq^q, -\6a^b^ pq.q^ +\6b^ P^q,q^ - 2A(xb^ P^q^q^ +\6aW p^q^q^ -Aa^q^q^ +20a^aq^q^ -32a^a^q^qj + 16aV^6<?7 + Ab^q^qT -IGab^q^qy +\6aWq^qT -I6b^ Pq^qj + AOod?^ Pq^q^ - l6a'b'Pq,q, -Sb'P'q.q, +l6ab'P\q, +Sl^q,q, -20al^q,q, +SaY,q,q, - Allq^q^ + 1 2cdlq^q^ - SaH^q^q^ + Al^q.q, - Sod^q^q^ + SbPq.q^q^ - 1 6abPq,q,q^ - Sbpq.q^q^ + 1 6abpq,q^q, + Wqlq, - 1 eoijqlq^ - SbPqjq^ + \6odjPqlq-, -Aa^aq^+Ua^a^q^ -SaWqT -Ab^q^ +\6ocb^qT -20a^b^q^ + Sa'b^q^ + l6b'Pq^ - 56cxb^pq^ + 56aWpq'^ - l6a'b^Pq^ - \2b''p\] + 144 40ab'fi'q^-36aVp'q^ +Sa'b'/3'q^ -4l^q^ +4odfq', +4cd^q', Sb^.q^ 4 l6(Xbfiq,q^ + AqW^ - %aq\q] - ^do^q^q] - Wq^] + 240*^,^7^ -\6a'bq,q^, + \6bJ3q,q^ - 40abfiq,q^ + I6a'bpq(,q', - 4a^bq, + lOa^obq, - 32a'a'bq, + \6aWbq^ - 4b^q^ + Socb^q^ + 4a^bl3q^ - 20a^0(bpq^ + 32aWbfiq^ - \6aWb/3q,+4b'^, -Sb'fi\+\6c(b'fi'q,+Sb'fi'q, -Sab'j3'q,+SbP,q, - lOocbl^q, + Sa^bl^q, - Sbfil^q, + 1 lodyfil^q, - Sa'b/3l^q, - 4bl^q, + 1 2ocbl^q, - ^a^bllq^ + 4bpllq^ -\2ab/M^,q, + Sa^bpl^q, + 4bllq^ - Sod^l^q, - 4bj3l^q, + SdyPl^q,+Sb'/3q,q,-\60(b'/3q,q,-ib'fi\q,+l6ab'l3'q,q, Sb'Pq.q, + \6c(b'/3q,q,+Sb'j3'q,q,-l6ab'fi\,q,+Sb\q,-\6ab^q,q, -Sb'j3q,q, + \6c(b'/3q^qs -Wq^q^+24oi}^q,q^-\6aWq,q^+24b''l5q,q^-64(Xb''pq,q^ + 32a^b^ I3q,q,-\6b^ P\-,q^+24oi)^ P^q,q,-\6aW P^.q^ -Sbq,q,q^ + \6(Xbq,q,q, + Sbj3q,q^q, -l6abPq,q,q, + Socbfiq^q, - 4b\l + %ab'ql + %b^ Pq] \6ab'fiq', -4b'P'ql +Sody'/3Vs) N.. = 4a\\-2ay ;» h (i-ay ^ 4a\\-2af /_ ^^,^ + ga V - 6aW + 2a'a' + 2b' /3' - 4cd,'/3' + 2a'b'/3' - If + 2od^ - a'lf - ll + 2odl - aHl - 2bpq, + \Qoi,pq, - ^a'b^, + a\l - 6abPq, + 6a'bJ3q, - 2aq,q, + q] - 4abJ3q, + 4a'bj3q, - 2094^6 + 2a'?496 + ^Qs^e ' 4«?5^6 + ql - 2aq\ + a'ql - 2bpq^ + Sc^yS^g - ba'bfiq^ - 2q,q^ + 2aqSi ) 1-a 13aV94 -12aV^4 +4a V^4 + 2b^fi'q,-6ab'P'q,+4aWfi'q,-l^q, + 2od^q, - a'lfq, - llq, + 4(Xllq, - 3a'l^q, - 2bPql + SO(bfiql - Sa'b/3ql + 2a'aq,-SaWq,+l0aWq, -4a'a'q,-2b'fi'q,+6ab'fi'q, -4aWj3\ + 145 ^50 2l^q, - Acd^q, + la'^l^s " '^U^ + ^OC^lq, + Abpq.q, - 1 %Cd}pq,q, + \Sa^bj3q,q, + 6oi?yQ^5^ - Sa^b/3q^^ + a^aq, - Sa^a^q^ + ^a^a'q^ - 4aWq^ - 2b'fi\ + 6ab'j3% - 4aVp'q, - Odfq, + a'lfq, + lljq, - 5od^q, + Sa'l^q, - lcdjl3q,q^ + Aa^bpq.q^ - la^bpq.q^ + loq^qsq^ " 4«'^495^6 " 2^596 + ^mU^ + Icdjpql - Aa^bpql - 2q,ql + 6aq,ql - Aa^.q] + a'q^ - Sa'oq^ + \OaWq, - lOflV^g +Aa^a'q, + 2b^/3\-6ab'P^q,+Aa^b^fi\, -lfq,+3od^q, - 2aHfq, - llq^ + 30dlq^ - 2aH^q, - Abfiq.q, + 1 6abPq,q, - \Aa^bfiq,q, - 2aqlq^ + Aa^qlq^ - ^abfiq^q^ + Sa^bfiq^q^ + 2q^q^q^ - Aoq.q^q^ + Ab/5q^q^ - \Aabfiqf,q^ + XAa^bpq^q^ + 2q,q^q^ - 6aq^q,q^ + Aa^q^q^q^ - 2bfiql + Sobfiq^ - %a'bPql) = -a^(a'-\Oa'a + AlaW -88aV +104aV -64aV +16aV +4aVy9' - 2Aa'ab'/3'+52aWb'j3' -ASa'a'b'j3' +l6aWb'fi' -2a'l^ + l2a'od^ - 26aWlf + 2Aa'a'lf - SaWl^ + /,' - 2od^ + aH^ - 2aHl + 12fl'< - 26aWll + 2Aa'aHl - Sa'a'l', - 2l^ll + Aod^ll - 2aH^ll + ll - 2al^ + a^ - Aa^bjSq, + 2Sa^0dj/3q, - 12aWbfiq, + ^OaWb^q, - 32aWbpq, + Ab/3lfq, - UocbplU, + Sa'bfil^q, - Abpljq, + \2od?pllq, - Sa^bJJl^q, + Al^qj -I6al^ql + \6a'liql -Aa'ab/3q, + 20aWbPq, - 32aWbfiq, + l6aWb/3q, + Aabfil^q, - Aa^bfilfq, -Aabpllq^ +Aa^bfil^q^+Aa^q^q,-2Sa^0iq,q^ +12aWq,qs - SOaWq^q, + 32a^a'q^q^ + Sb^fi^^q^ - 2Aoi}^ p^^q^ + I6a^b^fi^q,q, - Alfq,q,+I2al^q,q, -Sa^lfq^q^ -Al^q,q, + 20odiq,q,-2Aa'l^q,q, Sbfiqlq, + 32oi)pqlq,-32a^bpq]q, +Aa^aql-20aWql+32a^a^q^^ -IGaWq^ - Ab'j3'ql+\6ab'/3'q',-l2a'b'/3'ql+Alfq',-l2od^ql +Sa'l^qj -Aodlq] + %aHlql + 9,bpq,ql - A0abJ3q,qj + ASa'bjSq.qj + Sab/3ql - 1 6a^b/iql - Ab'fi'ql + leab'fiVe -l6aV/3'ql + Aljql - l6od',ql + \6aHlql - Aqlq] + \eaqlql-\6a^qlql -Aa^b^^ + 2Aa^abfiq^ -52a^a^bPq, +ASaWb/3q, - I6a^a*bfiq^ - Abfilfq^ + Socbfilfq^ - Aa^bj3lfq^ + Abfil^q, - ^obpllq^ + Aa'bfil^q, - Sb'fi'q.q, + 2Ac(b'fi\q, -\6aWj3\q, + Sbj3qlq, - 146 32cd)Pqlq,+32a-bJ3qlq,-Sab'fi'q,q,+Sa'b'fi^q,q,-Sbfiq,q,q, + 40(XbPq,q,q,-4Sa'bj3q,q,q, -Sabfiq^q, +\6a'bPqlq, +4a^q,q, -ISa^oq.q, 4 12aWq,q,-S0aWq,q,+32a^a'q^q,+^b^fi\q,-32ab^fi^q,q, + 32aWfi'q,q,-4lfq,q,+l2odfq,q,-SaH^q,q, -4l',q,q, +2Qodiq,q, - 24aHlq^q, - 8bfiq,q,q, + 32abfiq,q,q, - 32a'bJ3q,q,q, - S(XbPq,q,q, + \6a^b/3q,qf,q, + Sq.q.q^q^ - 32aq,q,q(^q^ + 32a\^q^qf,q^ + ^b^qlq^ - 32ab/3qlq, + 32a'bJ3qlq, + 4a^aql - 20aWql + 32aWql -\6a'a*q^ + Sab'PV,-\2a'b'0Vs +4iy,-l2ody,+Sa'lfq', -4odlql +Sa'l',q', + WPq.ql - 32oi}Pq,ql + 32a'bfiq,ql - 4qlql + \eaqlql - \6a'qlql + SabJ3q,q~ I6a'bj3q,ql -SbJ3q,ql +40abflq,ql -4Sa'bl3q,ql -Sab/3ql +\6a'bfiql) N.. = 16a' A^g3 = ^-^-^ib-2ab + a^b-2ab/3 + 2a^bJ3 + a^qs+oaiT-a^qT+oai^-a^q^) \i CX) N,,= ^{-2a' +Sa'a-lOaW +4aW -b'+2od?' -«V - 4b' J3' + Sod?' J3' - 4aV/3' + 2ll - 4cdl + 2a'll + 2ll - 4odl + la'll - 6bq, +]2abq, - 6a'bq, + Sotbfiq^ - Sa'bfiq^ - a'q] + 2oi)q^ - 2a'bq^ + 4bpq^ - Sabfiq^ + 4a'bpq^ - 60(q^q^ + 6a^9597 - <?7 + 2aq] - a'q] - 2oi)q^ + 2a'bq^ + 4bpq^ - Sabpq^ + 4a'b/3q^ - 2aq^q^ + 2a'q^q^ - 2q^q^ + 4aq^q^ - 2a'q^q^ - q] + 20^8 - a'ql ) N., =- \-a (- 2a'b + 6a'od? - 6a'a'b + 2a'a'b -b'+(xb'+ 4a'cd}fi - Sa'a'bfi + Aa'a^bp + bl] - cebl' - bll + cdyll + 2a' q^ - 6a'oeq^ + 6a'a'q^ - 2aWq^ + 3b' q, - 3ab'q, + 4b'P'q, - 4ceb'fi'q, - 3l^q, + 3al^q, - l^q, + Od^q, + 2bq', - 2abq' - 2abfiq^ - 2a'aq^ + 4a'a'q^ - 2a'a^q^ - ab'q^ + 2b' pq, - 2ab'/Jq^ + Odlq, - Odlq, - ebpq^q^ + 6cxb/3q,q^ + 2aqlq, + 2q^q'^ - 2aq^q^ - 2a'aq^ + 4a'a'q^ - 2a'a'q^ + CXb'q^ - 2b' pq^ + 2od?'pq, - od^q, + Od^q, - 2bfiq,q, + 2(Xbl3q^q^-2bq^q^+2od)q^q^+2q^q^q^-2aq,q^q^+2bql-2cd)ql) 147 N^ = -4a' + \6a'a - 24aW + 16a V - 4a V - 4a '^' + Sa'cxb' - 4aWb' - b' - Sa'b'fi'+l6a'ab'P' -SaWb'fi'+4a'l', -Sa'od^ +4aWl^ + 2b'l', -it + 4aHl - %a^0dl + 4aWll - Ib'll + lllll - it + 4b' q] + 4b'p'ql - 4l',q, + Sa'bPq, -I6a'ab/3q, +&aWbPq, +4b'/3q, -4b^lq, +4bpilq, Sbfiq^q, - 4iy, + 4qWi + ^a'bPq, - I6a'abj3q, + SaWbj^q, - 4b'fiq, + 4bfil^q, - 4bfil^q^ -8a'97^8 +16a'Q97^8 -%a'a'q^q^ -4b'q^q^+4l^q^q^+4l^q^q^ + 2^2 4bV,-^lks *■-►• J A ,:; ■- * . f LIST OF REFERENCES 1. Ball, R. S., A Treatise on the Theory of Screws, Cambridge University Press, London, 1900. 2. Buttolo, P., and Hannaford, B., "Advantages of Actuation Redundancy for the Design of Haptic Displays," Proceedings of ASME Fourth Annual Symposium on Haptic Interfaces for Virtual Environment and Teleoperator Systems, Vol. DSC 57-2, pp. 623-630, San Francisco, November 1995. 3. Buttolo, P., and Hannaford, B., "Pen-Based Force Display for Precision Manipulation in Virtual Environments," Proceedings of IEEE Virtual Reality Annual International Symposium, pp. 217-225, Raleigh, North Carolina, March 1995. 4. Cox, D. J., "The Dynamic Modeling and Command Signal Formulation for Parallel Multi-Parameter Robotic Devices," Master Thesis, Mechanical Engineering, University of Florida, 1981. 5. Crane, C, and Duffy, J., Kinematic Analysis of Robot Manipulators, Cambridge University Press, 1998. • . 6. Dasgupta, B., and Mruthyunjaya, T. S., "Force Redundancy in Parallel Manipulators: Theoretical and Practical Issues," Mechanism and Machine Theory, Vol. 33, No. 6, pp. 727-742, 1998. - , . . ,- 7. Dasgupta, B., and Mruthyunjaya, T. S., "The Stewart Platform Manipulator: A Review," Mechanism and Machine Theory, Vol. 35, No. 1, pp. 15-40, 2000. 8. Duffy, J., Statics and Kinematics with Applications to Robotics, Cambridge University Press, London, 1996. 9. Duffy, J., Rooney, J., Knight, B., and Crane, C, "A Review of a Family of Self- Deploying Tensegrity structures with Elastic Ties," The Shock and Vibration Digest, Vol. 32, No. 2, pp. 100-106, 2000. 10. Fichter, E. F., "A Stewart Platform-Based Manipulator: General Theory and Practical Construction," The International Journal of Robotics Research, Vol. 5, pp. 157-182, 1986. 148 149 11. 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Zhang, Y., Crane, C, and Duffy, J., "Determination of the Unique Orientation of Two Bodies Connected by a Ball-and-Socket Joint From Four Measured Displacements," Journal of Robotic Systems, Vol. 15, No. 5, pp. 299-308, 1998. -S :; { J BIOGRAPHICAL SKETCH Yu Zhang was bom on September 16, 1969, in Beijing, People's Republic of China. He received the Bachelor of Engineering degree in Mechanical Engineering from Dalian University of Technology in August 1992 and the Master of Engineering degree in Mechanical and Electronic Engineering from Beijing University of Posts and Telecommunications in April 1995. He began his Ph.D. study in Mechanical Engineering at the University of Florida in August 1996. 152 I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Josegji Duffy, Ch Graduate Research Pibfessor of Mechanical Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. r Carl D. Crane m Professor of Mechanical Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. u jLJg/VJ^ Gloria Jt,JA^iens Associate Professor of Mechanical Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. L\A^ Ali A. Seirig Ebaugh Professor of Mechanical Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. RalgltJSelfridj Professor of Compdter and Information Science and Engineering This dissertation was submitted to the Graduate Faculty of the College of Engineering and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. December 2000 r 0- M. Jack Ohanian Dean, College of Engineering Winfred M. Phillips Dean, Graduate School 1 i ,^- .■ 20 00 UNIVERSITY OF FLORIDA 3 1262 08555 1728